Hindawi Publishing CorporationISRN GeometryVolume 2013 Article ID 732432 12 pageshttpdxdoiorg1011552013732432

Research ArticleOn Twisted Products Finsler Manifolds

E Peyghan1 A Tayebi2 and L Nourmohammadi Far1

1 Faculty of Science Department of Mathematics Arak University Arak 38156-8-8349 Iran2 Faculty of Science Department of Mathematics Qom University Qom 3716146611 Iran

Correspondence should be addressed to L Nourmohammadi Far lnourmohammadigmailcom

Received 16 May 2013 Accepted 10 June 2013

Academic Editors I Biswas and A Borowiec

Copyright copy 2013 E Peyghan et al This is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

On the product of two Finsler manifolds1198721times119872

2 we consider the twisted metric F which is constructed by using Finsler metrics

1198651and 119865

2on the manifolds119872

1and119872

2 respectively We introduce horizontal and vertical distributions on twisted product Finsler

manifold and study C-reducible and semi-C-reducible properties of this manifold Then we obtain the Riemannian curvature andsome of non-Riemannian curvatures of the twisted product Finsler manifold such as Berwald curvature mean Berwald curvatureand we find the relations between these objects and their corresponding objects on119872

1and119872

2 Finally we study locally dually flat

twisted product Finsler manifold

1 IntroductionTwisted and warped product structures are widely used ingeometry to construct new examples of semi-Riemannianmanifolds with interesting curvature properties (see [1ndash3])Twisted productmetric tensors as a generalization of warpedproduct metric tensors have also been useful in the study ofseveral aspects of submanifold theory namely in hypersur-faces of complex space forms [4] in Lagrangian submanifolds[5] in decomposition of curvature netted hypersurfaces [6]and so forth

The notion of twisted product of Riemannian manifoldswas mentioned first by Chen in [7] and was generalizedfor the pseudo-Riemannian case by Ponge and Reckziegel[8] Chen extended the study of twisted product for CR-submanifolds in Kahler manifolds [9]

On the other hand Finsler geometry is a natural exten-sion of Riemannian geometry without the quadratic restric-tion Therefore it is natural to extend the construction oftwisted product manifolds for Finsler geometry In [10]Kozma-Peter-Shimada extended the construction of twistedproduct for the Finsler geometry

Let (1198721 119865

1) and (119872

2 119865

2) be two Finsler manifolds with

Finslermetrics1198651and119865

2 respectively and let119891 119872

1times119872

2rarr

119877+ be a smooth function On the product manifold119872

1times119872

2

we consider the metric

119865 (V1 V

2) = radic119865

2

1(V

1) + 1198912 (119909 119910) 119865

2

2(V

2) (1)

for all (119909 119910) isin 1198721times 119872

2and (V

1 V

2) isin 119879119872

∘

1times 119879119872

∘

2 where

119879119872∘

1is the slit tangent manifold 119879119872∘

1= 119879119872

1 ∘ The

manifold 1198721times 119872

2endowed with this metric we call the

twisted product of the manifolds1198721and119872

2and denote it by

1198721times119891119872

2 The function 119891 will be called the twisted function

In particular if 119891 is constant on1198722 then119872

1times119891119872

2is called

warped product manifoldLet (119872 119865) be a Finsler manifold The second and third

order derivatives of (12)1198652119909at 119910 isin 119879

119909119872

0are the symmetric

trilinear forms g119910and C

119910on 119879

119909119872 which called the fun-

damental tensor and Cartan torsion respectively A Finslermetric is called semi-C-reducible if its Cartan tensor is givenby

119862119894119895119896=119901

1 + 119899ℎ

119894119895119868119896+ ℎ

119895119896119868119894+ ℎ

119896119894119869119895 +119902

1198622119868119894119868119895119868119896 (2)

where 119901 = 119901(119909 119910) and 119902 = 119902(119909 119910) are scalar function on119879119872ℎ119894119895is the angular metric and 1198622 = 119868119894119868

119894[11] If 119902 = 0 then 119865

is called C-reducible Finsler metric and if 119901 = 0 then 119865 iscalled 1198622-like metric

The geodesic curves of a Finsler metric 119865 on a smoothmanifold 119872 are determined by the system of second-orderdifferential equations 119888119894 + 2119866119894( 119888) = 0 where the localfunctions 119866119894 = 119866119894(119909 119910) are called the spray coefficients 119865is called a Berwald metric if 119866119894 are quadratic in 119910 isin 119879

119909119872 for

any 119909 isin 119872 Taking a trace of Berwald curvature yields meanBerwald curvature E Then 119865 is said to be isotropic mean

2 ISRN Geometry

Berwaldmetric ifE = ((119899+1)2)119888119865minus1h whereh = ℎ119894119895119889119909

119894otimes119889119909

119895

is the angular metric and 119888 = 119888(119909) is a scalar function on119872[12]

The second variation of geodesics gives rise to a family oflinear maps R

119910= 119877

119894

119896119889119909

119896otimes (120597120597119909

119894)|119909 119879

119909119872 rarr 119879

119909119872 at

any point 119910 isin 119879119909119872 119877

119910is called the Riemann curvature in

the direction 119910 A Finsler metric 119865 is said to be of scalar flagcurvature if for some scalar functionK on119879119872

0the Riemann

curvature is in the form 119877119894119896= K1198652ℎ119894

119895 If K = constant then

119865 is said to be of constant flag curvatureIn this paper we introduce the horizontal and vertical

distributions on tangent bundle of a doubly warped productFinslermanifold and construct the Finsler connection on thismanifold Then we study some geometric properties of thisproduct manifold such as C-reducible and semi-C-reducibleThen we introduce the Riemmanian curvature of twistedproduct Finsler manifold (119872

1times119891119872

2 119865) and find the relation

between it and Riemmanian curvatures of its components(119872

1 119865

1) and (119872

2 119865

2) In the cases that (119872

1times119891119872

2 119865) is flat

or it has the scalar flag curvature we obtain some results onits components Then we study twisted product Finsler met-rics with vanishing Berwald curvature and isotropic meanBerwald curvature respectively Finally we study locallydually flat twisted product Finsler manifold We prove thatthere is not exist any locally dually flat proper twisted productFinsler manifold

2 Preliminary

Let 119872 be an 119899-dimensional 119862infin manifold Denote by 119879119909119872

the tangent space at 119909 isin 119872 by 119879119872 = cup119909isin119872119879119909119872 the tangent

bundle of119872 and by 119879119872∘= 119879119872 0 the slit tangent bundle

on119872 [13] A Finsler metric on119872 is a function 119865 119879119872 rarr

[0infin) which has the following properties

(i) 119865 is 119862infin on 119879119872∘(ii) 119865 is positively 1-homogeneous on the fibers of tangent

bundle 119879119872(iii) for each 119910 isin 119879

119909119872 the following quadratic form g

119910on

119879119909119872 is positive definite

g119910(119906 V) =

1

2

1205972

120597119904120597119905[119865

2(119910 + 119904119906 + 119905V)] |

119904119905=0 119906 V isin 119879

119909119872

(3)

Let 119909 isin 119872 and 119865119909= 119865|

119879119909119872 To measure the non-Euclidean

feature of 119865119909 define C

119910 119879

119909119872otimes 119879

119909119872otimes 119879

119909119872 rarr R by

C119910(119906 V 119908) =

1

2

119889

119889119905[g

119910+119905119908(119906 V)] |

119905=0 119906 V 119908 isin 119879

119909119872

(4)

The family C = C119910119910isin119879119872

∘ is called the Cartan torsion It iswell known that C = 0 if and only if 119865 is Riemannian [14]

For 119910 isin 119879119909119872

∘ define mean Cartan torsion I119910by I

119910(119906) =

119868119894(119910)119906

119894 where 119868119894= 119892

119895119896119862119894119895119896 119862

119894119895119896= (12)(120597119892

119894119895120597119910

119896) and 119906 =

119906119894(120597120597119909

119894)|119909 By Deickersquos theorem 119865 is Riemannian if and only

if I119910= 0

Let (119872 119865) be a Finsler manifold For 119910 isin 119879119909119872

∘ definethe Matsumoto torsion M

119910 119879

119909119872 otimes 119879

119909119872 otimes 119879

119909119872 rarr R by

M119910(119906 V 119908) = 119872

119894119895119896(119910)119906

119894V119895119908119896 where

119872119894119895119896= 119862

119894119895119896minus1

119899 + 1119868

119894ℎ119895119896+ 119868

119895ℎ119894119896+ 119868

119896ℎ119894119895 (5)

where ℎ119894119895= 119865119865

119910119894119910119895 is the angular metric In [15] it is proved

that a Finsler metric 119865 on a manifold119872 of dimension 119899 ge 3is a Randers metric if and only if M

119910= 0 for all 119910 isin 119879119872

0

A Randers metric 119865 = 120572 + 120573 on a manifold 119872 is just aRiemannian metric 120572 = radic119886119894119895119910119894119910119895 perturbed by a one form120573 = 119887

119894(119909)119910

119894 on119872 such that 120573120572lt 1

A Finsler metric is called semi-C-reducible if its Cartantensor is given by

119862119894119895119896=119901

1 + 119899ℎ

119894119895119868119896+ ℎ

119895119896119868119894+ ℎ

119896119894119868119895 +119902

1198622119868119894119868119895119868119896 (6)

where 119901 = 119901(119909 119910) and 119902 = 119902(119909 119910) are scalar function on119879119872 and 1198622 = 119868119894119868

119894with 119901 + 119902 = 1 In [11] Matsumoto-Shibata

proved that every (120572 120573)metric on amanifold119872 of dimension119899 ge 3 is semi-C-reducible

Given a Finslermanifold (119872 119865) then a global vector fieldG is induced by 119865 on 119879119872∘ which in a standard coordinate(119909

119894 119910

119894) for 119879119872∘ is given by G = 119910119894(120597120597119909119894) minus 2119866119894(119909 119910)(120597120597119910119894)

where

119866119894=1

411989211989411989712059721198652

120597119909119896120597119910119897119910119896minus120597119865

2

120597119909119897 119910 isin 119879

119909119872 (7)

G is called the spray associated to (119872 119865) In local coordinatesa curve 119888(119905) is a geodesic if and only if its coordinates (119888119894(119905))satisfy 119888119894 + 2119866119894( 119888) = 0 [16]

A Finslermetric119865 = 119865(119909 119910) on amanifold119872 is said to belocally dually flat if at any point there is a coordinate system(119909

119894) in which the spray coefficients are in the following form

119866119894= minus1

2119892119894119895119867

119910119895 (8)

where119867 = 119867(119909 119910) is a119862infin scalar function on119879119872∘ satisfying119867(119909 120582119910) = 120582

3119867(119909 119910) for all 120582 gt 0 Such a coordinate system

is called an adapted coordinate system In [17] Shen provedthat the Finsler metric 119865 on an open subset 119880 sub R119899 is duallyflat if and only if it satisfies (1198652)

119909119896119910119897119910

119896= 2(119865

2)119909119897

For a tangent vector 119910 isin 119879119909119872

∘ define B119910 119879

119909119872otimes119879

119909119872otimes

119879119909119872 rarr 119879

119909119872 and E

119910 119879

119909119872otimes 119879

119909119872 rarr R by B

119910(119906 V 119908) =

119861119894

119895119896119897(119910)119906

119895V119896119908119897(120597120597119909

119894)|119909and E

119910(119906 V) = 119864

119895119896(119910)119906

119895V119896 where

119861119894

119895119896119897=

1205973119866119894

120597119910119895120597119910119896120597119910119897 119864

119895119896=1

2119861119898

119895119896119898 (9)

B and E are called the Berwald curvature and mean Berwaldcurvature respectivelyThen 119865 is called a Berwaldmetric andweakly Berwald metric if B = 0 and E = 0 respectively [14]It is proved that on a Berwald space the parallel translationalong any geodesic preserves theMinkowski functionals [18]

ISRN Geometry 3

A Finsler metric 119865 is said to be isotropic Berwald metricand isotropic mean Berwald metric if its Berwald curvatureand mean Berwald curvature are in the following formrespectively

119861119894

119895119896119897= 119888 119865

119910119895119910119896120575

119894

119897+ 119865

119910119896119910119897120575

119894

119895+ 119865

119910119897119910119895120575

119894

119896+ 119865

119910119895119910119896119910119897119910

119894

119864119894119895=1

2(119899 + 1) 119888119865

minus1ℎ119894119895

(10)

where 119888 = 119888(119909) is a scalar function on119872 [19]The Riemann curvature R

119910= 119877

119894

119896119889119909

119896otimes (120597120597119909

119894)|119909

119879119909119872 rarr 119879

119909119872 is a family of linear maps on tangent spaces

defined by

119877119894

119896= 2120597119866

119894

120597119909119896minus 119910

119895 1205972119866119894

120597119909119895120597119910119896+ 2119866

119895 1205972119866119894

120597119910119895120597119910119896

minus120597119866

119894

120597119910119895

120597119866119895

120597119910119896

(11)

The flag curvature in Finsler geometry is a natural extensionof the sectional curvature in Riemannian geometry was firstintroduced by L Berwald [20] For a flag 119875 = span119910 119906 sub119879119909119872with flagpole119910 the flag curvatureK = K(119875 119910) is defined

by

K (119875 119910) =g119910(119906R

119910(119906))

g119910(119910 119910) g

119910(119906 119906) minus g

119910(119910 119906)

2 (12)

We say that a Finsler metric 119865 is of scalar curvature if for any119910 isin 119879

119909119872 the flag curvature K = K(119909 119910) is a scalar function

on the slit tangent bundle119879119872∘ IfK = constant then119865 is saidto be of constant flag curvature

3 Nonlinear Connection

Let (1198721 119865

1) and (119872

2 119865

2) be two Finsler manifolds Then the

functions

(i) 119892119894119895(119909 119910) =

1

2

12059721198652

1(119909 119910)

120597119910119894120597119910119895

(ii) 119892120572120573(119906 V) =

1

2

12059721198652

2(119906 V)

120597V120572120597V120573

(13)

define a Finsler tensor field of type (0 2) on 119879119872∘

1and

119879119872∘

2 respectively Now let (119872

1times119891119872

2 119865) be a doubly warped

Finsler manifold x = (119909 119906) isin 119872 y = (119910 V) isin 119879x119872119872 = 119872

1times 119872

2 and 119879x119872 = 1198791199091198721

oplus 119879119906119872

2 Then by using

(13) we conclude that

(g119886119887(119909 119906 119910 V)) = (

1

2

12059721198652(119909 119906 119910 V)

120597y119886y119887) = [

119892119894119895

0

0 1198912119892120572120573

]

(14)

where y119886 = (119910119894 V120572) g119894119895= 119892

119894119895 g

120572120573= 119891

2119892120572120573 g

119894120573= g

120572119895=

0 119894 119895 isin 1 1198991 120572 120573 isin 1 119899

2 and 119886 119887 isin

1 1198991+ 119899

2

Now we consider spray coefficients of 1198651 119865

2 and 119865 as

119866119894(119909 119910) =

1

4119892119894ℎ(12059721198652

1

120597119910ℎ120597119909119895119910119895minus120597119865

2

1

120597119909ℎ) (119909 119910) (15)

119866120572(119906 V) =

1

4119892120572120574(12059721198652

2

120597V120574120597119906120573V120573 minus

1205971198652

2

120597119906120574) (119906 V) (16)

G119886(x y) = 1

4g119886119887 ( 120597

21198652

120597y119887120597x119888y119888 minus 120597119865

2

120597x119887) (x y) (17)

Taking into account the homogeneity of both 11986521and 1198652

2

and using (15) and (16) we can conclude that 119866119894 and 119866120572are positively homogeneous of degree two with respect to(119910

119894) and (V120572) respectively Hence from Euler theorem for

homogeneous functions we infer that

120597119866119894

120597119910119895119910119895= 2119866

119894

120597119866120572

120597V120573V120573 = 2119866120572 (18)

By setting 119886 = 119894 in (17) we have

G119894(119909 119906 119910 V) =

1

4g119894ℎ ( 120597

21198652

120597119910ℎ120597119909119895119910119895+12059721198652

120597119910ℎ120597119906120572V120572 minus

1205971198652

120597119909ℎ)

(19)Direct calculations give us

1205971198652

120597119909ℎ=120597119865

2

1

120597119909ℎ+120597119891

2

120597119909ℎ1198652

2

12059721198652

120597119910ℎ120597119909119895=12059721198652

1

120597119910ℎ120597119909119895

12059721198652

120597119910ℎ120597119906120572= 0

(20)

Putting these equations together g119894ℎ = 119892119894ℎ in the previousequation and using (15) imply that

G119894(119909 119906 119910 V) = 119866119894 (119909 119910) minus

1

2119891119891

1198941198652

2 (21)

Similarly by setting 119886 = 120572 in (17) and using (16) we obtainG120572(119909 119906 119910 V) = 119866120572 (119906 V)

+ 119891minus1(119891

119895V120572119910119895 + 119891

120582V120572V120582 minus

1

21198911205741198921205721205741198652

2)

(22)

where 119891119894= 120597119891120597119909

119894 119891120574= 120597119891120597119906

120574 119891119894 = 119892119894ℎ119891ℎ and 119891120574 = 119892120582120574119891

120582

Therefore we have G119886= (G119894

G120572) where G119886 G119894 and G120572 are

given by (17) (21) and (22) respectivelyNow we put

(i) G119886

119887=120597G119886

120597y119887

(ii) 119866119894119895=120597119866

119894

120597119910119895

(iii) 119866120572120573=120597119866

120572

120597V120573

(23)

Then we have the following

4 ISRN Geometry

Lemma 1 The coefficients G119886

119887defined by (23) satisfy in the

following

(G119886

119887(119909 119906 119910 V)) = [

G119894

119895(119909 119906 119910 V) G120572

119895(119909 119906 119910 V)

G119894

120573(119909 119906 119910 V) G120572

120573(119909 119906 119910 V)] (24)

where

G119894

119895(119909 119906 119910 V) =

120597G119894

120597119910119895= 119866

119894

119895+ 119862

119894ℎ

119895119891119891

ℎ1198652

2 (25)

G119894

120573(119909 119906 119910 V) =

120597G119894

120597V120573= minus119891119891

119894V120573 (26)

G120572

119895(119909 119906 119910 V) =

120597G120572

120597119910119895= 119891

minus1119891119895V120572 (27)

G120572

120573(119909 119906 119910 V) =

120597G120572

120597V120573

= 119866120572

120573+ 119891

minus1(119862

120572120574

1205731198911205741198652

2+ 119891

119895119910119895120575120572

120573

minus 119891120572V

120573+ 119891

120573V120572 + 119891

120574V120574120575120572

120573)

(28)

Next 119881119879119872∘ kernel of the differential of the projectionmap

120587 = (1205871 120587

2) 119879119872

∘

1oplus 119879119872

∘

2997888rarr 119872

1times119872

2 (29)

which is a well-defined subbundle of 119879119879119872∘ is consid-ered Locally Γ(119881119879119872∘

) is spanned by the natural vectorfields 1205971205971199101 1205971205971199101198991 120597120597V1 120597120597V1198992 and it is calledthe twisted vertical distribution on 119879119872∘ Then using thefunctions given by (25)ndash(28) the nonholonomic vector fieldsare defined as follows

120575119905

120575119905119909119894=120597

120597119909119894minus G119895

119894

120597

120597119910119895minus G120573

119894

120597

120597V120573 (30)

120575119905

120575119905119906120572=120597

120597119906120572minus G119895

120572

120597

120597119910119895minus G120573

120572

120597

120597V120573 (31)

which make it possible to construct a complementary vectorsubbundle119867119879119872∘ to 119881119879119872∘ in 119879119879119872∘ as follows

119867119879119872∘= span 120575

119905

1205751199051199091

120575119905

1205751199051199091198991120575119905

1205751199051199061

120575119905

1205751199051199061198992 (32)

119867119879119872∘ is called the twisted horizontal distribution on 119879119872∘

Thus the tangent bundle of 119879119872∘ admits the decomposition

119879119879119872∘= 119867119879119872

∘oplus 119881119879119872

∘ (33)

It is shown thatG = (G119886

119887) is a nonlinear connection on119879119872 =

1198791198721oplus 119879119872

2 In the following we compute the nonlinear

connection of a twisted product Finsler manifold

Proposition 2 If (1198721times119891119872

2 119865) is a twisted product Finsler

manifold then G = (G119886

119887) is the nonlinear connection on 119879119872

Further one has

120597G119894

119895

120597119910119896119910119896+

120597G119894

119895

120597V120574V120574 = G119894

119895

120597G119894

120573

120597119910119896119910119896+

120597G119894

120573

120597V120574V120574 = G119894

120573

120597G120572

119895

120597119910119896119910119896+

120597G120572

119895

120597V120574V120574 = G120572

119895

120597G120572

120573

120597119910119896119910119896+

120597G120572

120573

120597V120574V120574 = G120572

120573

(34)

Definition 3 Using decomposition (33) the twisted verticalmorphism V119905 119879119879119872∘

rarr 119881119879119872∘ is defined by

V119905 =120597

120597119910119894otimes 120575

119905119910119894+120597

120597V120572otimes 120575

119905V120572 (35)

where

120575119905119910119894= 119889119910

119894+ G119894

119895119889119909

119895+ G119894

120573119889119906

120573

120575119905V120572 = 119889V120572 + G120572

119895119889119909

119895+ G120572

120573119889119906

120573

(36)

For this projective morphism the following hold

V119905 (120597

120597119910119894) =

120597

120597119910119894 V119905 (

120597

120597V120572) =

120597

120597V120572

V119905 (120575119905

120575119905119909119894) = 0 V119905 (

120575119905

120575119905119906119894) = 0

(37)

From the previous equations we conclude that

(V119905)2

= V119905 ker (V119905) = 119867119879119872∘ (38)

This mapping is called the twisted vertical projective

Definition 4 Using decomposition (33) the doubly warpedhorizontal projective ℎ119905 119879119879119872∘

rarr 119867119879119872∘ is defined by

ℎ119905= 119894119889 minus V119905 (39)

or

ℎ119905=120575119905

120575119905119909119894otimes 119889119909

119894+120575119905

120575119905119906120572otimes 119889119906

120572 (40)

For this projective morphism the following hold

ℎ119905(120575119905

120575119905119909119894) =

120575119905

120575119905119909119894 ℎ

119905(120575119905

120575119905119906120572) =

120575119905

120575119905119906120572

ℎ119905(120597

120597119910119894) = 0 ℎ

119905(120597

120597V120572) = 0

(41)

Thus we result that

(ℎ119905)2

= ℎ119905 ker (ℎ119905) = 119881119879119872∘

(42)

ISRN Geometry 5

Definition 5 Using decomposition (33) the twisted almosttangent structure 119869119905 119867119879119872∘

rarr 119881119879119872∘ is defined by

119869119905120597

120597119910119894otimes 119889119909

119894+120597

120597V120572otimes 119889119906

120572 (43)

or

119869119905(120575119905

120575119905119909119894) =

120597

120597119910119894 119869

119905(120575119905

120575119905119906120572) =

120597

120597V120572

119869119905(120597

120597119910119894) = 119869

119905(120597

120597V120572) = 0

(44)

Thus we result that

(119869119905)2

= 0 ker 119869119905 = 119868119898119869119905 = 119881119879119872∘ (45)

Here we introduce some geometrical objects of twistedproduct Finsler manifold In order to simplify the equationswe rewritten the basis of119867119879119872∘ and 119881119879119872∘ as follows

120575119905

120575119905x119886=120575119905

120575119905119909119894120575119894

119886+120575119905

120575119905119906120572120575120572

119886

120597

120597y119886=120597

120597119910119894120575119894

119886+120597

120597V120572120575120572

119886

(46)

Thus

119879119879119872∘= span 120575

119905

120575119905x119886120597

120597y119886 (47)

The Lie brackets of this basis is given by

[120575119905

120575119905x119886120575119905

120575119905x119887] = R119888

119886119887

120597

120597y119888

[120575119905

120575119905x119886120597

120597y119887] = G119888

119886119887

120597

120597y119888

[120597

120597y119886120597

120597y119887] = 0

(48)

where

(i) R119888

119886119887=120575119905G119888

119886

120575119905x119887minus120575119905G119888

119887

120575119905x119886 (49)

(ii) G119888

119886119887=120597G119888

119886

120597y119887 (50)

Therefore we have the following

Corollary 6 Let (1198721times119891119872

2 119865) be a twisted product Finsler

manifold Then

R119888

119886119887= (R119896

119894119895R119896

119894120573R119896

120572119895R119896

120572120573R120574

119894119895R120574

119894120573R120574

120572119895R120574

120572120573)

(51)

where

R119896

119894119895=120575119905G119896

119894

120575119905119909119895minus

120575119905G119896

119895

120575119905119909119894 R119896

119894120573=120575119905G119896

119894

120575119905119906120573minus

120575119905G119896

120573

120575119905119909119894

R119896

120572119895=120575119905G119896

120572

120575119905119909119895minus

120575119905G119896

119895

120575119905119906120572 R119896

120572120573=120575119905G119896

120572

120575119905119906120573minus

120575119905G119896

120573

120575119905119906120572

R120574

119894119895=120575119905G120574

119894

120575119905119909119895minus

120575119905G120574

119895

120575119905119909119894 R120574

119894120573=120575119905G120574

119894

120575119905119906120573minus

120575119905G120574

120573

120575119905119909119894

R120574

120572119895=120575119905G120574

120572

120575119905119909119895minus

120575119905G120574

119895

120575119905119906120572 R120574

120572120573=120575119905G120574

120572

120575119905119906120573minus

120575119905G120574

120573

120575119905119906120572

(52)

With a simple calculation we have the following

Corollary 7 Let (1198721times119891119872

2 119865) be a twisted product Finsler

manifold Then

G119888

119886119887= (G119896

119894119895G119896

119894120573G119896

120572119895G119896

120572120573G120574

119894119895G120574

119894120573G120574

120572119895G120574

120572120573) (53)

where

G120574

120572120573=120597G120574

120572

120597V120573

= 119866120574

120572120573+ 119891

minus1(119862

120574120582

1205721205731198911205821198652

2+ 2119862

120574120582

120572119891120582V120573+ 2119862

120574120582

120573119891120582V120572

minus 119891120574119892120572120573+ 119891

120573120575120574

120572+ 119891

120572120575120574

120573) = G120574

120573120572

G119896

119894119895=120597G119896

119894

120597119910119895= 119866

119896

119894119895+ 119862

119896ℎ

119894119895119891119891

ℎ1198652

2= G119896

119895119894

G119896

119894120573=120597G119896

119894

120597V120573= 2119862

119896ℎ

119894119891119891

ℎV120573= G119896

120573119894

G119896

120572120573=120597G119896

120572

120597V120573= minus119891119891

119896119892120572120573= G119896

120573120572

G120574

119894120573=120597G120574

119894

120597V120573= 119891

minus1119891119894120575120574

120573= G120574

120573119894

G120574

119894119895=120597G120574

119894

120597119910119895= G120574

119895119894= 0

(54)

where 119862119896ℎ119894119895= 120597119862

119896ℎ

119894120597119910

119895 Apart from G119888

119886119887 the functions F119888

119886119887are

given by

F119888119886119887=1

2g119888119890 (120575

119905g119890119886

120575119905x119887+120575119905g

119890119887

120575119905x119886minus120575119905g

119886119887

120575119905x119890) (55)

Corollary 8 Let (1198721times119891119872

2 119865) be a twisted product Finsler

manifold Then

F119888119886119887= (F119896

119894119895 F119896

119894120573 F119896

120572119895 F119896

120572120573 F120574

119894119895 F120574

119894120573 F120574

120572119895 F120574

120572120573) (56)

6 ISRN Geometry

where

F119896119894119895= 119865

119896

119894119895minus (119872

119903

119895119862119896

119894119903+119872

119903

119894119862119896

119895119903minus119872

119903

ℎ119862119894119895119903119892119896ℎ) (57)

F119896119894120573= minusG119903

120573119862119896

119894119903= F119896

120573119894 (58)

F119896120572120573= minus119891119891

119896119892120572120573+ 119891

2119892119896ℎG120582

ℎ119862120572120573120582 (59)

F120574119894119895= 119891

minus2119892120574120582G119903

120582119862119894119895119903 (60)

F120574119894120573= 119891

minus1119891119894120575120574

120573minus G120572

119894119862120574

120572120573= F120574

120573119894 (61)

F120574120572120573= 119865

120574

120572120573+ 119873

120574

120572120573minus (119872

120583

120573119862120574

120572120583+119872

120583

120572119862120574

120573120583minus119872

120583

120582119862120572120573120583119892120574120582)

(62)

119865119896

119894119895=1

2119892119896ℎ(120575119892

ℎ119894

120575119909119895+

120575119892ℎ119895

120575119909119894minus

120575119892119894119895

120575119909ℎ)

119865120574

120572120573=1

2119892120574120582(120575119892

120582120572

120575119906120573+

120575119892120582120573

120575119906120572minus

120575119892120572120573

120575119906120582)

119872119903

119894= 119862

119903ℎ

119894119891119891

ℎ1198652

2

119872120583

120572= 119891

minus1(119862

120583120574

1205721198911205741198652

2+ 119891

119903119910119903120575120583

120572+ 119891

120574V120574120575120583

120572minus 119892

120583120574119891120574V120572+ 119891

120572V120583)

119873120574

120572120573= 119891

minus1(119891

120573120575120574

120572+ 119891

120572120575120574

120573minus 119891

120582119892120574120582119892120572120573)

(63)

Proof By using (55) we have

F119896119894119895=1

2119892119896ℎ(120575119905119892ℎ119894

120575119905119909119895+

120575119905119892ℎ119895

120575119905119909119894minus

120575119905119892119894119895

120575119905119909ℎ) (64)

Since 119892119894119895is a function with respect to (119909 119910) then by (25) and

(30) we obtain

120575119905119892ℎ119894

120575119905119909119895=120575119892

ℎ119894

120575119909119895minus 2119872

119903

119895119862ℎ119894119903 (65)

Interchanging 119894 119895 and ℎ in the previous equation gives us

120575119905119892ℎ119895

120575119905119909119894=

120575119892ℎ119895

120575119909119894minus 2119872

119903

119894119862ℎ119895119903

120575119905119892119894119895

120575119905119909ℎ=

120575119892119894119895

120575119909ℎminus 2119872

119903

ℎ119862119894119895119903

(66)

Putting these equation in (64) give us (57) In the similar waywe can prove the another relation

By using (i) of (23) and (57)ndash(62) we can conclude thefollowing

Lemma 9 Let (1198721times119891119872

2 119865) be a twisted product Finsler

manifold Then y119888F119886119887119888= G119886

119887 where F119886

119887119888and G119886

119887are defined by

(55) and (i) of (23) respectively

The Cartan torsion is one of the most important non-Riemannian quantity in Finsler geometry and it is first

introduced by Finsler and emphasized by Cartan whichmeasures a departure from a Riemannian manifold Moreprecisely a Finsler metric reduces to a Riemannian metricif and only if it has vanishing Cartan torsion The localcomponents of Cartan tensor field of the twisted Finslermanifold (119872

1times119891119872

2 119865) is defined by

C119886

119887119888=1

2g119886119890 120597g119887119890120597y119888 (67)

From this definition we conclude the following

Lemma 10 Let119862119896119894119895and119862120574

120572120573be the local components of Cartan

tensor field on1198721and119872

2 respectively Then one has

C119888

119886119887= (C119896

119894119895C119896

119894120573C119896

120572119895C119896

120572120573C120574

119894119895C120574

119894120573C120574

120572119895C120574

120572120573) (68)

where

C119896

119894119895=1

2119892119896ℎ120597119892

119894119895

120597119910ℎ= 119862

119896

119894119895

C120574

120572120573=1

2119892120574120582120597119892

120572120573

120597V120582= 119862

120574

120572120573

(69)

and C119896

119894120573= C119896

120572119895= C119896

120572120573= C120574

119894119895= C120574

119894120573= C120574

120572119895= 0

By using the Lemma 10 we can get the following

Corollary 11 Let (1198721times119891119872

2 119865) be a twisted product Finsler

manifoldThen (1198721times119891119872

2 119865) is a Riemannianmanifold if and

only if (1198721 119865

1) and (119872

2 119865

2) are Riemannian manifold

Various interesting special forms of Cartan tensors havebeen obtained by some Finslerians [11] The Finsler spaceshaving such special forms have been called C-reducible C2-like semi-C-reducible and so forth In [21] Matsumotointroduced the notion of C-reducible Finsler metrics andproved that any Randers metric is C-reducible Later onMatsumoto-Hojo proves that the converse is true too [15]

Here we define the Matsumoto twisted tensorM119886119887119888

for atwisted product Finsler manifold (119872

1times119891119872

2 119865) as follows

M119886119887119888= C

119886119887119888minus1

119899 + 1I

119886h119887119888+ I

119887h119886119888+ I

119888h119886119887 (70)

where I119886= g119887119888C

119886119887119888C

119886119887119888= g

119888119889C119889

119886119887 andh

119886119887= g

119886119887minus(1119865

2)y

119886y119887

By attention to the previous equation and relations

C119894119895119896= 119862

119894119895119896 C

120572120573120574= 119891

2119862120572120573120574 (71)

we obtain

M120572119895119896= minus

1

119899 + 1119868

120572(119892

119895119896minus1

1198652119910119895119910119896) minus1198912

1198652V120572(119868

119895119910119896+ 119868

119896119910119895)

(72)

Contracting the previous equation in 119910119895119910119896 gives us

119910119895119910119896M

120572119895119896= minus11989121198652

11198652

2

(119899 + 1) 1198652119868120572 (73)

ISRN Geometry 7

Similarly we obtain

V120582V120573M119894120573120582= minus11989121198652

11198652

2

(119899 + 1) 1198652119868119894 (74)

Therefore if M119894120573120582= M

120572119895119896= 0 then we get 119868

119894= 119868

120572= 0 that

is (1198721 119865

1) and (119872

2 119865

2) are Riemannian manifolds Thus we

have the following

Theorem 12 There is not exist any C-reducible twisted prod-uct Finsler manifold

Now we are going to consider semi-C-reducible twistedproduct Finsler manifold (119872

1times119891119872

2 119865) Let (119872

1times119891119872

2 119865) be

a semi-C-reducible twisted product Finsler manifold Thenwe have

C119886119887119888=119901

119899 + 1I

119886h119887119888+ I

119887h119886119888+ I

119888h119886119887 +119902

C2I119886I119887I119888 (75)

where C2= I119886I

119886and 119901 and 119902 are scalar function on119872

1times119891119872

2

with 119901 + 119902 = 1 This equation gives us

0 = C120572119895119896

=119901

119899 + 1119868

120572(119892

119895119896minus1

1198652119910119895119910119896) minus1198912

1198652V120572(119868

119895119910119896+ 119868

119896119910119895)

+119902

C2119868120572119868119895119868119896

(76)

Contractiing the previous equation with 119910119895119910119896 implies that

11990111989121198652

11198652

2119868120572= 0 (77)

Therefore we have 119901 = 0 or 119868120572= 0 If 119901 = 0 then

119865 is 1198622-like metric But if 119901 = 0 then 119868120572= 0 that is

1198652is Riemannian metric In this case with similar way

we conclude that 1198651is Riemannian metric But definition

119865 cannot be a Riemannian metric Therefore we have thefollowing

Theorem 13 Every semi-C-reducible twisted product Finslermanifold (119872

1times119891119872

2 119865) is a 1198622-like manifold

4 Riemannian Curvature

The Riemannian curvature of twisted product Finsler man-ifold (119872

1times119891119872

2 119865) with respect to Berwald connection is

given by

R 119886

119887 119888119889=120575119905F119886

119887119888

120575119905x119889minus120575119905F119886

119887119889

120575119905x119888+ F119886

119889119890F119890119887119888minus F119886

119888119890F119890119887119889 (78)

Lemma 14 Let (1198721times119891119872

2 119865) be a twisted product Finsler

manifold Then one has

R119886

119888119889= y119887R 119886

119887 119888119889 (79)

where R119886

119888119889and y119887R 119886

119887 119888119889are given by (50) and (78)

Proof By using (78) we have

y119887R 119894

119887 119896119897= y119887120575119905F119894

119887119896

120575119905x119897minus y119887120575119905F119894

119887119897

120575119905x119896+ y119887F119894

119897119890F119890119887119896 minus y119887F119894

119896119890F119890119887119897 (80)

By using Corollary 8 and Lemma 9 we obtain

y119887120575119905F119894

119887119896

120575119905x119897=120575119905G119894

119896

120575119905119909119897+ F119894

119895119896G119895

119897+ F119894

120573119896G120573

119897

y119887F119894119897119890F119890119887119896= F119894

119897ℎGℎ

119896+ F119894

119897120574G120574

119896

(81)

Interchanging 119894 and 119895 in the previous equation implies that

y119887120575119905F119894

119887119897

120575119905x119896=120575119905G119894

119897

120575119905119909119896+ F119894

119895119897G119895

119896+ F119894

120573119897G120573

119896

y119887F119894119896119890F119890119887119897= F119894

119896ℎGℎ

119897+ F119894

119896120574G120574

119897

(82)

Setting (81) and (82) in (80) gives us y119887R119894

119887 119896119897= R119894

119896119897 In the

similar way we can obtain this relation for another indices

Using (78) we can compute the Riemannian curvature ofa twisted product Finsler manifold

Lemma 15 Let (1198721times119891119872

2 119865) be a twisted product Finsler

manifoldThen the coefficients of Riemannian curvature are asfollows

R119894

119895 119896119897= 119877

119894

119895 119896119897

minus 119872119903

119897

120597119865119894

119895119896

120597119910119903+

120575119905119872

119894

119895119896

120575119905119909119897+119865

119894

119897ℎ119872

ℎ

119895119896+119872

119894

119897ℎ119865ℎ

119895119896minus119872

119894

119897ℎ119872

ℎ

119895119896

+ 119891minus2119892120572120574G119903

120572G119898

120574119862119894

119897119903119862119895119896119898 minusC

119896

119897

(83)

R 119894

120572 119896119897= minus

120575119905

120575119905119909119897(G119903

120572119862119894

119896119903) minus (119865

119894

119903119897minus119872

119894

119903119897)G119898

120572119862119903

119896119898

minus119891minus1G119903

120573119862119894

119897119903119891119896120575120573

120572+ G119903

120573G120583

119896119862119894

119897119903119862120573

120572120583 minus C

119896

119897

(84)

R 119894

119895 120573120582= minus

120575119905

120575119905119906120582(G119903

120573119862119894

119895119903) + G119898

120582G119897

120573119862119894

119903119898119862119903

119895119897

minus (119891119894119892120572120582minus 119891G120583

ℎ119892119894ℎ119862120572120582120583) (119891

119895120575120572

120573minus 119891G]

119895119862120572

120573])

minus C120573

120582

(85)

8 ISRN Geometry

R 119894

120572 120573119897=120575119905

120575119905119906120573(G119903

120572119862119894

119897119903) minus120575119905

120575119905119909119897119891 (119891

119894119892120572120573minus 119891G120582

ℎ119892119894ℎ119862120572120573120582)

minus G119898

120573G119904

120572119862119894

119903119898119862119903

119897119904+ (119891

119894119892120583120573minus 119891G120582

ℎ119892119894ℎ119862120583120573120582)

times (119891119897120575120583

120572minus 119891G]

119897119862120583

120572]) minus 119891119892119903ℎ(119865

119894

119903119897minus119872

119894

119903119897)

times (119891ℎ119892120572120573minus119891G120582

ℎ119862120572120573120582)minusG119903

120583119862119894

119897119903(119865

120583

120572120573+ 119873

120583

120572120573minus119872

120583

120572120573)

(86)

R 119894

119895 120573119897= minus

120575119905

120575119905119909119897(G119903

120573119862119894

119895119903) minus

120575119905

120575119905119906120573(119865

119894

119895119897minus119872

119894

119895119897)

minus (119865119894

119897119903minus119872

119894

119897119903)G119904

120573119862119903

119895119904minus 119891

minus1G119903

120572119862119894

119897119903

times (119891119895120575120572

120573minus 119891G120583

119895119862120572

120573120583) + G119904

120573119862119894

119903119904(119865

119903

119895119897minus119872

119903

119895119897)

+ 119891minus1G119903

120583119862119895119897119903(119891

119894120575120583

120573minus 119891G120582

ℎ119892119894ℎ119862120583

120573120582)

(87)

R 119894

120572 120573120582= minus

120575119905

120575119905119906120582(119891119891

119894119892120572120573minus 119891

2119892119894ℎG120583

ℎ119862120572120573120583) + 119891G119904

120582119862119894

119903119904

times(119891119903119892120572120573minus119891G120583

119897119862120572120573120583119892119903119897)minus119891 (119891

119894119892120582120583minus119891119892

119894ℎG120581

ℎ119862120582120583120581)

times (119865120583

120572120573+ 119873

120583

120572120573minus119872

120583

120572120573) minus C

120573

120582

(88)

R 120574

119895 119896119897=

120575119905

120575119905119909119897(119891

minus2119892120574120582G119903

120582119862119895119896119903) + 119891

minus2119892120574120582G119904

120582119862119897119903119904

times (119865119903

119895119896minus119872

119903

119895119896) + 119891

minus3G119903

120583119862119895119896119903(119891

119897119892120574120583minus 119891G120572

119897119862120574120583

120572)

minus C119896

119897

(89)

R 120574

119895 120573119897=120575119905

120575119905119909119897(119891

minus1119891119895120575120574

120573minus 119891G120572

119895119862120574

120572120573)

minus120575119905

120575119905119906120573(119891

minus2119892120574120582G119903

120582119862119895119897119903) minus 119891

minus2119892120574120582G119904

120582G119898

120573119862ℎ

119897119904119862ℎ119895119898

+ 119891minus2(119891

119897120575120574

120583minus 119891G120572

119897119862120574

120583120572) (119891

119895120575120583

120573minus 119891G]

119895119862120583

120573])

minus 119891minus1(119891

119903120575120574

120573minus 119891G120572

119903119862120574

120573120572) (119865

119903

119895119897minus119872

119903

119895119897)

minus 119891minus2119892120583120582G119903

120582119862119895119897119903(119865

120574

120573120583+ 119873

120574

120573120583minus119872

120574

120573120583)

(90)

R 120574

120572 120573119897=120575119905

120575119905119909119897(119865

120574

120572120573+ 119873

120574

120572120573minus119872

120574

120572120573)

minus120575119905

120575119905119906120573(119891

minus1119891119897120575120574

120572minus G120583

119897119862120574

120572120583)

minus 119891minus1119892120574120582G119904

120582119862ℎ

119897119904(119891

ℎ119892120572120573minus 119891G120583

ℎ119862120572120573120583)

+ 119891minus1(119891

119897120575120574

120583minus 119891G120581

119897119862120574

120583120581) (119865

120583

120572120573+ 119873

120583

120572120573minus119872

120583

120572120573)

+ 119891minus1G119904

120572119862119903

119897119904(119891

119903120575120574

120573minus 119891G120581

119903119862120574

120573120581)

minus 119891minus1(119891

119897120575120583

120572minus 119891G120581

119897119862120583

120572120581) (119865

120574

120573120583+ 119873

120574

120573120583minus119872

120574

120573120583)

(91)

R 120574

119895 120573120582=

120575119905

120575119905119906120582(119891

minus1119891119895120575120574

120573minus G120572

119895119862120574

120572120573)

+ 119891minus1(119865

120574

120572120582+ 119873

120574

120572120582minus119872

120574

120572120582) (119891

119895120575120572

120573minus 119891G]

119895119862120572

120573])

minus119891minus1G119898

120573119862119903

119895119898(119891

119903120575120574

120582minus 119891G120572

119903119862120574

120582120572) minus C

120573

120582

(92)

R 120574

120572 119896119897=

120575119905

120575119905119909119897(119891

minus1119891119896120575120574

120572minus G120583

119896119862120574

120572120583)

+ 119891minus2(119891

119897120575120574

120573minus 119891G120581

119897119862120574

120573120581) (119891

119896120575120573

120572minus 119891G]

119896119862120573

120572])

minus119891minus2G119904

120583G119898

120572119892120574120583119862ℎ

119897119904119862ℎ119896119898 minus C

119896

119897

(93)

R 120574

120572 120573120582= 119877

120574

120572 120573120582minus 119872

120581

120582

120597119865120574

120572120573

120597V120581+

120575119905119872

120574

120572120573

120575119905119906120582+119865

120574

120582120583119872

120583

120572120573+119872

120574

120582120583119865120583

120572120573

minus119872120574

120582120583119872

120583

120572120573+120575119905119873

120574

120572120582

120575119905119906120573+ 119865

120574

120573120583119873

120583

120572120582

+ 119873120574

120573120583119865120583

120572120582+ 119873

120574

120573120583119873

120583

120572120582+ 119873

120574

120582120583119872

120583

120572120573

+ 119873120583

120572120573119872

120574

120582120583+ (119892

119903119904120575120574

120582119891119904minus 119891119892

119903119904G120581

119904119862120574

120582120581)

times (119892120572120573119891119903minus 119891G120583

119903119862120572120573120583) minusC

120573

120582

(94)

where

119872119894

119895119896= 119872

119903

119896119862119894

119895119903+119872

119903

119895119862119894

119896119903minus119872

119903

ℎ119892119894ℎ119862119895119896119903

119872120574

120572120573= 119872

120583

120573119862120574

120572120583+119872

120583

120572119862120574

120573120583minus119872

120583

] 119892120574]119862120572120573120583

119873120574

120572120573= 119891

minus1(119891

120573120575120574

120572+ 119891

120572120575120574

120573minus 119891

120582119892120574120582119892120572120573)

(95)

and C119894

119895denotes the interchange of indices 119894 119895 and subtraction

ByTheorem 18 we have the following

Theorem 16 Let (1198721times119891119872

2 119865) be a flat twisted product

Finsler manifold and let (1198721 119865

1) be Riemannian If 119891 is a

function on1198722 only then (119872

1 119865

1) is locally flat

Similarly we get the following

Theorem 17 Let (1198721times119891119872

2 119865) be a flat twisted product

Finsler manifold and let (1198722 119865

2) be Riemannian If 119891 is a

function on 1198721 only then (119872

2 119865

2) is a space of positive

constant curvature ||119892119903119886119889119891||2

ISRN Geometry 9

Proof Since 1198722is Riemannain and 119891 is a function on 119872

1

then by (94) we obtain

R 120574

120572 120573120582= 119877

120574

120572 120573120582+1003817100381710038171003817119892119903119886119889119891

1003817100381710038171003817

2

(120575120574

120582119892120572120573minus 120575

120574

120573119892120572120582) (96)

Since (1198721times119891119872

2 119865) is flat then R120574

120572 120573120582= 0 Thus the proof is

complete

Theorem 18 Let (1198721times119891119872

2 119865) be a twisted product Rieman-

nian manifold and let 119891 be a function on 1198722 only Then

(1198721times119891119872

2 119865) is flat if and only if (119872

1 119865

1) is flat and the

Riemannian curvature of (1198722 119865

2) satisfies in the following

equation

119877120574

120572 120573120582= 120575119905119873

120574

120572120582

120575119905119906120573+ 119865

120574

120573120583119873

120583

120572120582+ 119873

120574

120573120583119865120583

120572120582+ 119873

120574

120573120583119873

120583

120572120582 minus C

120573

120582

(97)

5 Twisted Product Finsler Manifolds withNon-Riemannian Curvature Properties

There are several important non-Riemannian quantities suchas the Berwald curvature B the mean Berwald curvatureE and the Landsberg curvature L [22] They all vanishfor Riemannian metrics hence they are said to be non-Riemannian In this section we find some necessary and suf-ficient conditions underwhich a twisted product Riemannianmanifold are Berwaldian of isotropic Berwald curvatureof isotropic mean Berwald curvature First we prove thefollowing

Lemma 19 Let (1198721times119891119872

2 119865) be a twisted product Finsler

manifold Then the coefficients of Berwald curvature are asfollows

B120574

120572120573120582= 119861

120574

120572120573120582+ 119891

minus1(119862

120574]120582120572120573119891]119865

2

2+ 2119862

120574]120572120573119891]V120582

+ 2119862120574]120572120582119891]V120573 + 2119862

120574]120572119891]119892120582120573

+ 2119862120574]120582120573119891]V120572 + 2119862

120574]120573119891]119892120582120572

+ 2119862120574]120582119891]119892120572120573 minus 2119862120572120573120582119891

120574)

(98)

B119896

119894119895119897= 119861

119896

119894119895119897+ 119891119862

119896ℎ

119897119895119894119891ℎ1198652

2 (99)

B119896

119894120573119897= 2119891119862

119896ℎ

119894119897119891ℎV120573 (100)

B119896

120572120573119897= 2119891119892

120572120573119862119896ℎ

119897119891ℎ (101)

B119896

120572120573120582= minus 2119891119862

120572120573120582119891119896 (102)

B120574

119894120573120582= B120574

119894119895120582= B120574

119894119895119896= 0 (103)

Let (1198721times119891119872

2 119865) is a Berwald manifold Then we have

B119889

119886119887119888= 0 By using (102) we get

119862120572120573120582119891119896= 0 (104)

Multiplying this equation in 119892119896119903 we obtain

119862120572120573120582119891119903= 0 (105)

Thus if 119891 is not constant on1198721 then we have 119862

120572120573120582= 0 Also

from (101) we result that

119862119896ℎ

119897119891ℎ= 0 (106)

Differentiating this equation with respect to 119910119895 gives us

119862119896ℎ

119897119895119891ℎ= 0 (107)

Similarly we obtain

119862119896ℎ

119897119895119894119891ℎ= 0 (108)

Setting the last equation in (99) implies that 119861119896119894119895119897= 0 that

is (1198721 119865

1) is Berwaldian These explanations give us the

following theorem

Theorem 20 Let (1198721times119891119872

2 119865) be a twisted product Finsler

manifold and let119891 be not constant on1198721Then (119872

1times119891119872

2 119865)

is Berwaldian if and only if (1198721 119865

1) is Berwaldian (119872

2 119865

2) is

Riemannian and the equation 119862119896ℎ119897119891ℎ= 0 is hold

But if 119891 is constant on1198721 that is 119891

119894= 0 then we get the

following

Theorem 21 Let (1198721times119891119872

2 119865) be a twisted product Finsler

manifold and 119891 is constant on 1198721 Then (119872

1times119891119872

2 119865)

is Berwaldian if and only if (1198721 119865

1) is Berwaldian and

the Berwald curvature of (1198722 119865

2) satisfies in the following

equation

119861120574

120572120573120582= minus 119891

minus1(119862

120574]120573120572120582119891]119865

2

2+ 2119862

120574]120573120572119891]V120582 + 2119862

120574]120582120572119891]V120573

+ 2119862120574]120572119891]119892120582120573 + 2119862

120574]120573120582119891]V120572

+ 2119862120574]120573119891]119892120582120572 + 2119862

120574]120582119891]119892120572120573

minus 2119892120574]119862120572120573120582119891])

(109)

Here we consider twisted product Finsler manifold(119872

1times119891119872

2 119865) of isotropic Berwald curvature

Theorem 22 Every isotropic Berwald twisted product Finslermanifold (119872

1times119891119872

2 119865) is a Berwald manifold

Proof Let (1198721times119891119872

2 119865) be an isotropic Berwald manifold

Then we have

B119889

119886119887119888= 119888119865

minus1h119889

119886h119887119888+ h119889

119887h119886119888+ h119889

119888h119886119887+ 2C

119886119887119888y119889 (110)

where 119888 = 119888(x) is a function on119872 Setting 119886 = 119895 119887 = 119896 119888 = 119897and 119889 = 120574 and using (103) imply that

119888119865minus13

1198652119910119895119910119896119910119897V120574 minus V120574 (119910

119895119892119896119897+ 119910

119896119892119895119897+ 119910

119897119892119895119896) = 0 (111)

Multiplying the previous equation in 119910119895119910119896 we derive that119888119891

21198652

11198652

2= 0 Thus we have 119888 = 0 that is (119872

1times119891119872

2) is

Berwaldian

10 ISRN Geometry

Now we are going to study twisted product Finslermanifold of isotropicmean Berwald curvature For this workwe must compute the coefficients of mean Berwald curvatureof a twisted product Finsler manifold

Lemma 23 Let (1198721times119891119872

2 119865) be a twisted product Finsler

manifold Then the coefficients of mean Berwald curvature areas follows

E120572120573= 119864

120572120573+ 119891119892

120572120573119868ℎ119891ℎ+1

2119891119868

]120572120573119891]119865

2

2+ 119891

minus1119891]

times (119862120574]120572120573

V120574+ 119868

]120572V120573+ 119868

]120573V120572+ 119862

]120572120573+ 119868

]119892120572120573)

(112)

E119894119895= 119864

119894119895+1

2119891119868

ℎ

119895119894119891ℎ1198652

2 (113)

E119894120573= 119891119868

ℎ

119894119891ℎV120573 (114)

where 119864119894119895

and 119864120572120573

are the coefficients of mean Berwaldcurvature of (119872

1 119865

1) and (119872

2 119865

2) respectively

Proof By definition and Lemma 19 we get the proof

Theorem24 The twisted product Finslermanifold (1198721times119891119872

2

119865) is weakly Berwald if and only if (1198721 119865

1) is weakly Berwald

119868ℎ119891ℎ= 0 and the following hold

119864120572120573= minus

1

2119891119868

]120572120573119891]119865

2

2minus 119891

minus1119891]

times (119862120574]120572120573

V120574+ 119868

]120572V120573+ 119868

]120573V120572+ 119862

]120572120573+ 119868

]119892120572120573)

(115)

Proof If (1198721times119891119872

2) be a weakly Berwald manifold then we

have

E120572120573= E

119894119895= E

119894120573= 0 (116)

Thus by using (114) we result that 119868ℎ119894119891ℎ= 0 This equation

implies that

119868ℎ

119895119894119891ℎ= 0 119868

ℎ119891ℎ= 0 (117)

By setting these equations in (112) and (113) we conclude that119864119894119895= 0 and 119864

120572120573satisfies in (115)

Now if 119891 is constant on1198722 then (115) implies that 119864

120572120573=

0 Thus we conclude the following

Corollary 25 Let (1198721times119891119872

2 119865) be a twisted product Finsler

manifold and let 119891 be a function on 1198721 only Then

(1198721times119891119872

2 119865) is weakly Berwald if and only if (119872

1 119865

1) and

(1198722 119865

2) are weakly Berwald manifolds and 119868ℎ119891

ℎ= 0

Now we consider twisted product Finsler manifolds withisotropic mean Berwald curvature It is remarkable that as aconsequence of Lemma 23 we have the following

Lemma26 Twisted product Finslermanifold (1198721times119891119872

2 119865) is

isotropic mean Berwald manifold if and only if

119864120572120573+ 119891119892

120572120573119868ℎ119891ℎ+1

2119891119868

]120572120573119891]119865

2

2+ 119891

minus1119891]

times (119862120574]120572120573

V120574+ 119868

]120572V120573+ 119868

]120573V120572+ 119862

]120572120573+ 119868

]119892120572120573)

minus119899 + 1

2119888119891

2119865minus1(119892

120572120573minus1198912

1198652V120572V120573) = 0

(118)

119864119894119895+1

2119891119868

ℎ

119895119894119891ℎ1198652

2minus119899 + 1

2119888119865

minus1(119892

119894119895minus1

1198652119910119894119910119895) = 0 (119)

119888 (119899 + 1) 119865minus3119910119894+ 119891119868

ℎ

119894119891ℎ= 0 (120)

where 119888 = 119888(x) is a scalar function on119872

Theorem 27 Every twisted product Finsler manifold(119872

1times119891119872

2 119865) with isotropic mean Berwald curvature is a

weakly Berwald manifold

Proof Suppose that 119865 is isotropic mean Berwald twistedproduct Finsler metric Then differentiating (120) withrespect to V120574 gives us

119888 (119899 + 1) 1198912119865minus5V

120574119910119894= 0 (121)

Thus we conclude that 119888 = 0 This implies that 119865 reduces to aweakly Berwald metric

6 Locally Dually Flat Twisted ProductFinsler Manifolds

In [23] Amari and Nagaoka introduced the notion of duallyflat Riemannian metrics when they study the informationgeometry on Riemannian manifolds Information geometryhas emerged from investigating the geometrical structure ofa family of probability distributions and has been appliedsuccessfully to various areas including statistical inferencecontrol system theory andmultiterminal information theoryIn Finsler geometry Shen extends the notion of locally duallyflatness for Finsler metrics [17] Dually flat Finsler metricsform a special and valuable class of Finsler metrics in Finslerinformation geometry which play a very important role instudying flat Finsler information structure [24 25]

In this section we study locally dually flat twisted productFinsler metrics It is remarkable that a Finsler metric 119865 =119865(x y) on a manifold119872 is said to be locally dually flat if atany point there is a standard coordinate system (x119886 y119886) in119879119872such that it satisfies

12059721198652

120597x119887120597y119886y119887 = 2120597119865

2

120597x119886 (122)

In this case the coordinate (x119886) is called an adapted localcoordinate system Byusing (122) we can obtain the followinglemma

ISRN Geometry 11

Lemma 28 Let (1198721times119891119872

2 119865) be a twisted product Finsler

manifold Then 119865 is locally dually flat if and only if 1198651and 119865

2

satisfy in the following equations

12059721198652

1

120597119909119896120597119910119897119910119896= 2120597119865

2

1

120597119909119897+ 4119891119891

1198971198652

2 (123)

4119891119896V120573119910119896+ 119891

12059721198652

2

120597119906120572120597V120573V120572 + 4119891

120572V120573V120572 = 2119891

1205971198652

2

120597119906120573+ 4119891

1205731198652

2

(124)

Now let 119865 be a locally dually flat Finsler metric Takingderivative with respect to V120574 from (123) yields 119891

119897= 0 which

means that 119891 is a constant function on1198721 In this case the

relations (123) and (124) reduce to the following

12059721198652

1

120597119909119896120597119910119897119910119896= 2120597119865

2

1

120597119909119897 (125)

11989112059721198652

2

120597119906120572120597V120573V120572 + 4119891

120572V120573V120572 = 2119891

1205971198652

2

120597119906120573+ 4119891

1205731198652

2 (126)

By (125) we deduce that 1198651is locally dually flat

Now we assume that 1198651and 119865

2are locally dually flat

Finsler metrics Then we have

12059721198652

1

120597119909119896120597119910119897119910119896= 2120597119865

2

1

120597119909119897

12059721198652

2

120597119906120572120597V120573V120572 = 2

1205971198652

2

120597119906120573

(127)

By (127) we derive that (123) and (124) are hold if and only ifthe following hold

119891119897= 0 119891

120572V120573V120572 = 119891

1205731198652

2 (128)

Therefore we can conclude the following

Theorem 29 Let (1198721times119891119872

2 119865) be a twisted product Finsler

manifold

(i) If 119865 is locally dually flat then 1198651is locally dually flat 119891

is a function with respect to (119906120572) only and 1198652satisfies

in (126)(ii) If 119865

1and 119865

2are locally dually flat then 119865 is locally

dually flat if and only if 119891 is a function with respect(119906

120572) only and 119865

2satisfies in (128)

ByTheorem 29 we conclude the following

Corollary 30 There is not exist any locally dually flat propertwisted product Finsler manifold

References

[1] J K Beem P E Ehrlich and K L Easley Global LorentzianGeometry vol 202 of Monographs and Textbooks in Pure andApplied Mathematics Marcel Dekker New York NY USA 2ndedition 1996

[2] G Ganchev and V Mihova ldquoRiemannian manifolds of quasi-constant sectional curvaturesrdquo Journal fur die Reine und Ange-wandte Mathematik vol 522 pp 119ndash141 2000

[3] B OrsquoNeill Semi-Riemannian Geometry With Applications toRelativity vol 103 of Pure and Applied Mathematics AcademicPress New York NY USA 1983

[4] M Lohnherr and H Reckziegel ldquoOn ruled real hypersurfacesin complex space formsrdquo Geometriae Dedicata vol 74 no 3pp 267ndash286 1999

[5] B-Y Chen F Dillen L Verstraelen and L VranckenldquoLagrangian isometric immersions of a real-space-form119872119899

(119888)

into a complex-space-form119872119899(4119888)rdquoMathematical Proceedings

of the Cambridge Philosophical Society vol 124 no 1 pp 107ndash125 1998

[6] N Koike ldquoThe decomposition of curvature netted hypersur-facesrdquo Geometriae Dedicata vol 54 no 1 pp 1ndash11 1995

[7] B-y Chen Geometry of Submanifolds and its ApplicationsScience University of Tokyo Tokyo Japan 1981

[8] R Ponge and H Reckziegel ldquoTwisted products in pseudo-Riemannian geometryrdquo Geometriae Dedicata vol 48 no 1 pp15ndash25 1993

[9] B-Y Chen ldquoTwisted product CR-submanifolds in Kaehlermanifoldsrdquo Tamsui Oxford Journal of Mathematical Sciencesvol 16 no 2 pp 105ndash121 2000

[10] L Kozma I R Peter and H Shimada ldquoOn the twisted productof Finsler manifoldsrdquo Reports on Mathematical Physics vol 57no 3 pp 375ndash383 2006

[11] M Matsumoto and C Shibata ldquoOn semi-C-reducibility T-tensor= 0 and S4-likeness of Finsler spacesrdquo Journal of Math-ematics of Kyoto University vol 19 no 2 pp 301ndash314 1979

[12] B Najafi Z Shen and A Tayebi ldquoFinsler metrics of scalar flagcurvature with special non-Riemannian curvature propertiesrdquoGeometriae Dedicata vol 131 pp 87ndash97 2008

[13] E Peyghan andAHeydari ldquoConformal vector fields on tangentbundle of a Riemannian manifoldrdquo Journal of MathematicalAnalysis and Applications vol 347 no 1 pp 136ndash142 2008

[14] Z Shen Differential Geometry of Spray and Finsler SpacesKluwer Academic Publishers Dordrecht The Netherlands2001

[15] M Matsumoto and S-I Hojo ldquoA conclusive theorem on C-reducible Finsler spacesrdquo The Tensor Society vol 32 no 2 pp225ndash230 1978

[16] D Bao S-S Chern and Z Shen An Introduction to Riemann-Finsler Geometry vol 200 of Graduate Texts in MathematicsSpringer New York NY USA 2000

[17] Z Shen ldquoRiemann-Finsler geometry with applications to infor-mation geometryrdquo Chinese Annals of Mathematics B vol 27 no1 pp 73ndash94 2006

[18] Y Ichijyo ldquoFinsler manifolds modeled on a Minkowski spacerdquoJournal of Mathematics of Kyoto University vol 16 no 3 pp639ndash652 1976

[19] X Chen and Z Shen ldquoOn Douglas metricsrdquo PublicationesMathematicae Debrecen vol 66 no 3-4 pp 503ndash512 2005

[20] L Berwald ldquoUber Parallelubertragung in Raumen mit allge-meiner Massbestimmungrdquo Jahresbericht der DMVmdashDeutscheMathematiker-Vereinigung vol 34 pp 213ndash220 1926

[21] M Matsumoto ldquoOn Finsler spaces with Randersrsquo metric andspecial forms of important tensorsrdquo Journal of Mathematics ofKyoto University vol 14 pp 477ndash498 1974

12 ISRN Geometry

[22] A Tayebi and E Peyghan ldquoOn special Berwald metricsrdquoSymmetry Integrability and Geometry vol 6 article 008 9pages 2010

[23] S-I Amari and H NagaokaMethods of Information Geometryvol 191 of Translations of Mathematical Monographs AmericanMathematical Society Providence RI USA 2000

[24] X Cheng Z Shen and Y Zhou ldquoOn locally dually flat Finslermetricsrdquo International Journal ofMathematics vol 21 no 11 pp1531ndash1543 2010

[25] Q Xia ldquoOn a class of locally dually flat Finsler metrics ofisotropic flag curvaturerdquo Publicationes Mathematicae Debrecenvol 78 no 1 pp 169ndash190 2011

Submit your manuscripts athttpwwwhindawicom

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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

2 ISRN Geometry

Berwaldmetric ifE = ((119899+1)2)119888119865minus1h whereh = ℎ119894119895119889119909

119894otimes119889119909

119895

is the angular metric and 119888 = 119888(119909) is a scalar function on119872[12]

The second variation of geodesics gives rise to a family oflinear maps R

119910= 119877

119894

119896119889119909

119896otimes (120597120597119909

119894)|119909 119879

119909119872 rarr 119879

119909119872 at

any point 119910 isin 119879119909119872 119877

119910is called the Riemann curvature in

the direction 119910 A Finsler metric 119865 is said to be of scalar flagcurvature if for some scalar functionK on119879119872

0the Riemann

curvature is in the form 119877119894119896= K1198652ℎ119894

119895 If K = constant then

119865 is said to be of constant flag curvatureIn this paper we introduce the horizontal and vertical

distributions on tangent bundle of a doubly warped productFinslermanifold and construct the Finsler connection on thismanifold Then we study some geometric properties of thisproduct manifold such as C-reducible and semi-C-reducibleThen we introduce the Riemmanian curvature of twistedproduct Finsler manifold (119872

1times119891119872

2 119865) and find the relation

between it and Riemmanian curvatures of its components(119872

1 119865

1) and (119872

2 119865

2) In the cases that (119872

1times119891119872

2 119865) is flat

or it has the scalar flag curvature we obtain some results onits components Then we study twisted product Finsler met-rics with vanishing Berwald curvature and isotropic meanBerwald curvature respectively Finally we study locallydually flat twisted product Finsler manifold We prove thatthere is not exist any locally dually flat proper twisted productFinsler manifold

2 Preliminary

Let 119872 be an 119899-dimensional 119862infin manifold Denote by 119879119909119872

the tangent space at 119909 isin 119872 by 119879119872 = cup119909isin119872119879119909119872 the tangent

bundle of119872 and by 119879119872∘= 119879119872 0 the slit tangent bundle

on119872 [13] A Finsler metric on119872 is a function 119865 119879119872 rarr

[0infin) which has the following properties

(i) 119865 is 119862infin on 119879119872∘(ii) 119865 is positively 1-homogeneous on the fibers of tangent

bundle 119879119872(iii) for each 119910 isin 119879

119909119872 the following quadratic form g

119910on

119879119909119872 is positive definite

g119910(119906 V) =

1

2

1205972

120597119904120597119905[119865

2(119910 + 119904119906 + 119905V)] |

119904119905=0 119906 V isin 119879

119909119872

(3)

Let 119909 isin 119872 and 119865119909= 119865|

119879119909119872 To measure the non-Euclidean

feature of 119865119909 define C

119910 119879

119909119872otimes 119879

119909119872otimes 119879

119909119872 rarr R by

C119910(119906 V 119908) =

1

2

119889

119889119905[g

119910+119905119908(119906 V)] |

119905=0 119906 V 119908 isin 119879

119909119872

(4)

The family C = C119910119910isin119879119872

∘ is called the Cartan torsion It iswell known that C = 0 if and only if 119865 is Riemannian [14]

For 119910 isin 119879119909119872

∘ define mean Cartan torsion I119910by I

119910(119906) =

119868119894(119910)119906

119894 where 119868119894= 119892

119895119896119862119894119895119896 119862

119894119895119896= (12)(120597119892

119894119895120597119910

119896) and 119906 =

119906119894(120597120597119909

119894)|119909 By Deickersquos theorem 119865 is Riemannian if and only

if I119910= 0

Let (119872 119865) be a Finsler manifold For 119910 isin 119879119909119872

∘ definethe Matsumoto torsion M

119910 119879

119909119872 otimes 119879

119909119872 otimes 119879

119909119872 rarr R by

M119910(119906 V 119908) = 119872

119894119895119896(119910)119906

119894V119895119908119896 where

119872119894119895119896= 119862

119894119895119896minus1

119899 + 1119868

119894ℎ119895119896+ 119868

119895ℎ119894119896+ 119868

119896ℎ119894119895 (5)

where ℎ119894119895= 119865119865

119910119894119910119895 is the angular metric In [15] it is proved

that a Finsler metric 119865 on a manifold119872 of dimension 119899 ge 3is a Randers metric if and only if M

119910= 0 for all 119910 isin 119879119872

0

A Randers metric 119865 = 120572 + 120573 on a manifold 119872 is just aRiemannian metric 120572 = radic119886119894119895119910119894119910119895 perturbed by a one form120573 = 119887

119894(119909)119910

119894 on119872 such that 120573120572lt 1

A Finsler metric is called semi-C-reducible if its Cartantensor is given by

119862119894119895119896=119901

1 + 119899ℎ

119894119895119868119896+ ℎ

119895119896119868119894+ ℎ

119896119894119868119895 +119902

1198622119868119894119868119895119868119896 (6)

where 119901 = 119901(119909 119910) and 119902 = 119902(119909 119910) are scalar function on119879119872 and 1198622 = 119868119894119868

119894with 119901 + 119902 = 1 In [11] Matsumoto-Shibata

proved that every (120572 120573)metric on amanifold119872 of dimension119899 ge 3 is semi-C-reducible

Given a Finslermanifold (119872 119865) then a global vector fieldG is induced by 119865 on 119879119872∘ which in a standard coordinate(119909

119894 119910

119894) for 119879119872∘ is given by G = 119910119894(120597120597119909119894) minus 2119866119894(119909 119910)(120597120597119910119894)

where

119866119894=1

411989211989411989712059721198652

120597119909119896120597119910119897119910119896minus120597119865

2

120597119909119897 119910 isin 119879

119909119872 (7)

G is called the spray associated to (119872 119865) In local coordinatesa curve 119888(119905) is a geodesic if and only if its coordinates (119888119894(119905))satisfy 119888119894 + 2119866119894( 119888) = 0 [16]

A Finslermetric119865 = 119865(119909 119910) on amanifold119872 is said to belocally dually flat if at any point there is a coordinate system(119909

119894) in which the spray coefficients are in the following form

119866119894= minus1

2119892119894119895119867

119910119895 (8)

where119867 = 119867(119909 119910) is a119862infin scalar function on119879119872∘ satisfying119867(119909 120582119910) = 120582

3119867(119909 119910) for all 120582 gt 0 Such a coordinate system

is called an adapted coordinate system In [17] Shen provedthat the Finsler metric 119865 on an open subset 119880 sub R119899 is duallyflat if and only if it satisfies (1198652)

119909119896119910119897119910

119896= 2(119865

2)119909119897

For a tangent vector 119910 isin 119879119909119872

∘ define B119910 119879

119909119872otimes119879

119909119872otimes

119879119909119872 rarr 119879

119909119872 and E

119910 119879

119909119872otimes 119879

119909119872 rarr R by B

119910(119906 V 119908) =

119861119894

119895119896119897(119910)119906

119895V119896119908119897(120597120597119909

119894)|119909and E

119910(119906 V) = 119864

119895119896(119910)119906

119895V119896 where

119861119894

119895119896119897=

1205973119866119894

120597119910119895120597119910119896120597119910119897 119864

119895119896=1

2119861119898

119895119896119898 (9)

B and E are called the Berwald curvature and mean Berwaldcurvature respectivelyThen 119865 is called a Berwaldmetric andweakly Berwald metric if B = 0 and E = 0 respectively [14]It is proved that on a Berwald space the parallel translationalong any geodesic preserves theMinkowski functionals [18]

ISRN Geometry 3

A Finsler metric 119865 is said to be isotropic Berwald metricand isotropic mean Berwald metric if its Berwald curvatureand mean Berwald curvature are in the following formrespectively

119861119894

119895119896119897= 119888 119865

119910119895119910119896120575

119894

119897+ 119865

119910119896119910119897120575

119894

119895+ 119865

119910119897119910119895120575

119894

119896+ 119865

119910119895119910119896119910119897119910

119894

119864119894119895=1

2(119899 + 1) 119888119865

minus1ℎ119894119895

(10)

where 119888 = 119888(119909) is a scalar function on119872 [19]The Riemann curvature R

119910= 119877

119894

119896119889119909

119896otimes (120597120597119909

119894)|119909

119879119909119872 rarr 119879

119909119872 is a family of linear maps on tangent spaces

defined by

119877119894

119896= 2120597119866

119894

120597119909119896minus 119910

119895 1205972119866119894

120597119909119895120597119910119896+ 2119866

119895 1205972119866119894

120597119910119895120597119910119896

minus120597119866

119894

120597119910119895

120597119866119895

120597119910119896

(11)

The flag curvature in Finsler geometry is a natural extensionof the sectional curvature in Riemannian geometry was firstintroduced by L Berwald [20] For a flag 119875 = span119910 119906 sub119879119909119872with flagpole119910 the flag curvatureK = K(119875 119910) is defined

by

K (119875 119910) =g119910(119906R

119910(119906))

g119910(119910 119910) g

119910(119906 119906) minus g

119910(119910 119906)

2 (12)

We say that a Finsler metric 119865 is of scalar curvature if for any119910 isin 119879

119909119872 the flag curvature K = K(119909 119910) is a scalar function

on the slit tangent bundle119879119872∘ IfK = constant then119865 is saidto be of constant flag curvature

3 Nonlinear Connection

Let (1198721 119865

1) and (119872

2 119865

2) be two Finsler manifolds Then the

functions

(i) 119892119894119895(119909 119910) =

1

2

12059721198652

1(119909 119910)

120597119910119894120597119910119895

(ii) 119892120572120573(119906 V) =

1

2

12059721198652

2(119906 V)

120597V120572120597V120573

(13)

define a Finsler tensor field of type (0 2) on 119879119872∘

1and

119879119872∘

2 respectively Now let (119872

1times119891119872

2 119865) be a doubly warped

Finsler manifold x = (119909 119906) isin 119872 y = (119910 V) isin 119879x119872119872 = 119872

1times 119872

2 and 119879x119872 = 1198791199091198721

oplus 119879119906119872

2 Then by using

(13) we conclude that

(g119886119887(119909 119906 119910 V)) = (

1

2

12059721198652(119909 119906 119910 V)

120597y119886y119887) = [

119892119894119895

0

0 1198912119892120572120573

]

(14)

where y119886 = (119910119894 V120572) g119894119895= 119892

119894119895 g

120572120573= 119891

2119892120572120573 g

119894120573= g

120572119895=

0 119894 119895 isin 1 1198991 120572 120573 isin 1 119899

2 and 119886 119887 isin

1 1198991+ 119899

2

Now we consider spray coefficients of 1198651 119865

2 and 119865 as

119866119894(119909 119910) =

1

4119892119894ℎ(12059721198652

1

120597119910ℎ120597119909119895119910119895minus120597119865

2

1

120597119909ℎ) (119909 119910) (15)

119866120572(119906 V) =

1

4119892120572120574(12059721198652

2

120597V120574120597119906120573V120573 minus

1205971198652

2

120597119906120574) (119906 V) (16)

G119886(x y) = 1

4g119886119887 ( 120597

21198652

120597y119887120597x119888y119888 minus 120597119865

2

120597x119887) (x y) (17)

Taking into account the homogeneity of both 11986521and 1198652

2

and using (15) and (16) we can conclude that 119866119894 and 119866120572are positively homogeneous of degree two with respect to(119910

119894) and (V120572) respectively Hence from Euler theorem for

homogeneous functions we infer that

120597119866119894

120597119910119895119910119895= 2119866

119894

120597119866120572

120597V120573V120573 = 2119866120572 (18)

By setting 119886 = 119894 in (17) we have

G119894(119909 119906 119910 V) =

1

4g119894ℎ ( 120597

21198652

120597119910ℎ120597119909119895119910119895+12059721198652

120597119910ℎ120597119906120572V120572 minus

1205971198652

120597119909ℎ)

(19)Direct calculations give us

1205971198652

120597119909ℎ=120597119865

2

1

120597119909ℎ+120597119891

2

120597119909ℎ1198652

2

12059721198652

120597119910ℎ120597119909119895=12059721198652

1

120597119910ℎ120597119909119895

12059721198652

120597119910ℎ120597119906120572= 0

(20)

Putting these equations together g119894ℎ = 119892119894ℎ in the previousequation and using (15) imply that

G119894(119909 119906 119910 V) = 119866119894 (119909 119910) minus

1

2119891119891

1198941198652

2 (21)

Similarly by setting 119886 = 120572 in (17) and using (16) we obtainG120572(119909 119906 119910 V) = 119866120572 (119906 V)

+ 119891minus1(119891

119895V120572119910119895 + 119891

120582V120572V120582 minus

1

21198911205741198921205721205741198652

2)

(22)

where 119891119894= 120597119891120597119909

119894 119891120574= 120597119891120597119906

120574 119891119894 = 119892119894ℎ119891ℎ and 119891120574 = 119892120582120574119891

120582

Therefore we have G119886= (G119894

G120572) where G119886 G119894 and G120572 are

given by (17) (21) and (22) respectivelyNow we put

(i) G119886

119887=120597G119886

120597y119887

(ii) 119866119894119895=120597119866

119894

120597119910119895

(iii) 119866120572120573=120597119866

120572

120597V120573

(23)

Then we have the following

4 ISRN Geometry

Lemma 1 The coefficients G119886

119887defined by (23) satisfy in the

following

(G119886

119887(119909 119906 119910 V)) = [

G119894

119895(119909 119906 119910 V) G120572

119895(119909 119906 119910 V)

G119894

120573(119909 119906 119910 V) G120572

120573(119909 119906 119910 V)] (24)

where

G119894

119895(119909 119906 119910 V) =

120597G119894

120597119910119895= 119866

119894

119895+ 119862

119894ℎ

119895119891119891

ℎ1198652

2 (25)

G119894

120573(119909 119906 119910 V) =

120597G119894

120597V120573= minus119891119891

119894V120573 (26)

G120572

119895(119909 119906 119910 V) =

120597G120572

120597119910119895= 119891

minus1119891119895V120572 (27)

G120572

120573(119909 119906 119910 V) =

120597G120572

120597V120573

= 119866120572

120573+ 119891

minus1(119862

120572120574

1205731198911205741198652

2+ 119891

119895119910119895120575120572

120573

minus 119891120572V

120573+ 119891

120573V120572 + 119891

120574V120574120575120572

120573)

(28)

Next 119881119879119872∘ kernel of the differential of the projectionmap

120587 = (1205871 120587

2) 119879119872

∘

1oplus 119879119872

∘

2997888rarr 119872

1times119872

2 (29)

which is a well-defined subbundle of 119879119879119872∘ is consid-ered Locally Γ(119881119879119872∘

) is spanned by the natural vectorfields 1205971205971199101 1205971205971199101198991 120597120597V1 120597120597V1198992 and it is calledthe twisted vertical distribution on 119879119872∘ Then using thefunctions given by (25)ndash(28) the nonholonomic vector fieldsare defined as follows

120575119905

120575119905119909119894=120597

120597119909119894minus G119895

119894

120597

120597119910119895minus G120573

119894

120597

120597V120573 (30)

120575119905

120575119905119906120572=120597

120597119906120572minus G119895

120572

120597

120597119910119895minus G120573

120572

120597

120597V120573 (31)

which make it possible to construct a complementary vectorsubbundle119867119879119872∘ to 119881119879119872∘ in 119879119879119872∘ as follows

119867119879119872∘= span 120575

119905

1205751199051199091

120575119905

1205751199051199091198991120575119905

1205751199051199061

120575119905

1205751199051199061198992 (32)

119867119879119872∘ is called the twisted horizontal distribution on 119879119872∘

Thus the tangent bundle of 119879119872∘ admits the decomposition

119879119879119872∘= 119867119879119872

∘oplus 119881119879119872

∘ (33)

It is shown thatG = (G119886

119887) is a nonlinear connection on119879119872 =

1198791198721oplus 119879119872

2 In the following we compute the nonlinear

connection of a twisted product Finsler manifold

Proposition 2 If (1198721times119891119872

2 119865) is a twisted product Finsler

manifold then G = (G119886

119887) is the nonlinear connection on 119879119872

Further one has

120597G119894

119895

120597119910119896119910119896+

120597G119894

119895

120597V120574V120574 = G119894

119895

120597G119894

120573

120597119910119896119910119896+

120597G119894

120573

120597V120574V120574 = G119894

120573

120597G120572

119895

120597119910119896119910119896+

120597G120572

119895

120597V120574V120574 = G120572

119895

120597G120572

120573

120597119910119896119910119896+

120597G120572

120573

120597V120574V120574 = G120572

120573

(34)

Definition 3 Using decomposition (33) the twisted verticalmorphism V119905 119879119879119872∘

rarr 119881119879119872∘ is defined by

V119905 =120597

120597119910119894otimes 120575

119905119910119894+120597

120597V120572otimes 120575

119905V120572 (35)

where

120575119905119910119894= 119889119910

119894+ G119894

119895119889119909

119895+ G119894

120573119889119906

120573

120575119905V120572 = 119889V120572 + G120572

119895119889119909

119895+ G120572

120573119889119906

120573

(36)

For this projective morphism the following hold

V119905 (120597

120597119910119894) =

120597

120597119910119894 V119905 (

120597

120597V120572) =

120597

120597V120572

V119905 (120575119905

120575119905119909119894) = 0 V119905 (

120575119905

120575119905119906119894) = 0

(37)

From the previous equations we conclude that

(V119905)2

= V119905 ker (V119905) = 119867119879119872∘ (38)

This mapping is called the twisted vertical projective

Definition 4 Using decomposition (33) the doubly warpedhorizontal projective ℎ119905 119879119879119872∘

rarr 119867119879119872∘ is defined by

ℎ119905= 119894119889 minus V119905 (39)

or

ℎ119905=120575119905

120575119905119909119894otimes 119889119909

119894+120575119905

120575119905119906120572otimes 119889119906

120572 (40)

For this projective morphism the following hold

ℎ119905(120575119905

120575119905119909119894) =

120575119905

120575119905119909119894 ℎ

119905(120575119905

120575119905119906120572) =

120575119905

120575119905119906120572

ℎ119905(120597

120597119910119894) = 0 ℎ

119905(120597

120597V120572) = 0

(41)

Thus we result that

(ℎ119905)2

= ℎ119905 ker (ℎ119905) = 119881119879119872∘

(42)

ISRN Geometry 5

Definition 5 Using decomposition (33) the twisted almosttangent structure 119869119905 119867119879119872∘

rarr 119881119879119872∘ is defined by

119869119905120597

120597119910119894otimes 119889119909

119894+120597

120597V120572otimes 119889119906

120572 (43)

or

119869119905(120575119905

120575119905119909119894) =

120597

120597119910119894 119869

119905(120575119905

120575119905119906120572) =

120597

120597V120572

119869119905(120597

120597119910119894) = 119869

119905(120597

120597V120572) = 0

(44)

Thus we result that

(119869119905)2

= 0 ker 119869119905 = 119868119898119869119905 = 119881119879119872∘ (45)

Here we introduce some geometrical objects of twistedproduct Finsler manifold In order to simplify the equationswe rewritten the basis of119867119879119872∘ and 119881119879119872∘ as follows

120575119905

120575119905x119886=120575119905

120575119905119909119894120575119894

119886+120575119905

120575119905119906120572120575120572

119886

120597

120597y119886=120597

120597119910119894120575119894

119886+120597

120597V120572120575120572

119886

(46)

Thus

119879119879119872∘= span 120575

119905

120575119905x119886120597

120597y119886 (47)

The Lie brackets of this basis is given by

[120575119905

120575119905x119886120575119905

120575119905x119887] = R119888

119886119887

120597

120597y119888

[120575119905

120575119905x119886120597

120597y119887] = G119888

119886119887

120597

120597y119888

[120597

120597y119886120597

120597y119887] = 0

(48)

where

(i) R119888

119886119887=120575119905G119888

119886

120575119905x119887minus120575119905G119888

119887

120575119905x119886 (49)

(ii) G119888

119886119887=120597G119888

119886

120597y119887 (50)

Therefore we have the following

Corollary 6 Let (1198721times119891119872

2 119865) be a twisted product Finsler

manifold Then

R119888

119886119887= (R119896

119894119895R119896

119894120573R119896

120572119895R119896

120572120573R120574

119894119895R120574

119894120573R120574

120572119895R120574

120572120573)

(51)

where

R119896

119894119895=120575119905G119896

119894

120575119905119909119895minus

120575119905G119896

119895

120575119905119909119894 R119896

119894120573=120575119905G119896

119894

120575119905119906120573minus

120575119905G119896

120573

120575119905119909119894

R119896

120572119895=120575119905G119896

120572

120575119905119909119895minus

120575119905G119896

119895

120575119905119906120572 R119896

120572120573=120575119905G119896

120572

120575119905119906120573minus

120575119905G119896

120573

120575119905119906120572

R120574

119894119895=120575119905G120574

119894

120575119905119909119895minus

120575119905G120574

119895

120575119905119909119894 R120574

119894120573=120575119905G120574

119894

120575119905119906120573minus

120575119905G120574

120573

120575119905119909119894

R120574

120572119895=120575119905G120574

120572

120575119905119909119895minus

120575119905G120574

119895

120575119905119906120572 R120574

120572120573=120575119905G120574

120572

120575119905119906120573minus

120575119905G120574

120573

120575119905119906120572

(52)

With a simple calculation we have the following

Corollary 7 Let (1198721times119891119872

2 119865) be a twisted product Finsler

manifold Then

G119888

119886119887= (G119896

119894119895G119896

119894120573G119896

120572119895G119896

120572120573G120574

119894119895G120574

119894120573G120574

120572119895G120574

120572120573) (53)

where

G120574

120572120573=120597G120574

120572

120597V120573

= 119866120574

120572120573+ 119891

minus1(119862

120574120582

1205721205731198911205821198652

2+ 2119862

120574120582

120572119891120582V120573+ 2119862

120574120582

120573119891120582V120572

minus 119891120574119892120572120573+ 119891

120573120575120574

120572+ 119891

120572120575120574

120573) = G120574

120573120572

G119896

119894119895=120597G119896

119894

120597119910119895= 119866

119896

119894119895+ 119862

119896ℎ

119894119895119891119891

ℎ1198652

2= G119896

119895119894

G119896

119894120573=120597G119896

119894

120597V120573= 2119862

119896ℎ

119894119891119891

ℎV120573= G119896

120573119894

G119896

120572120573=120597G119896

120572

120597V120573= minus119891119891

119896119892120572120573= G119896

120573120572

G120574

119894120573=120597G120574

119894

120597V120573= 119891

minus1119891119894120575120574

120573= G120574

120573119894

G120574

119894119895=120597G120574

119894

120597119910119895= G120574

119895119894= 0

(54)

where 119862119896ℎ119894119895= 120597119862

119896ℎ

119894120597119910

119895 Apart from G119888

119886119887 the functions F119888

119886119887are

given by

F119888119886119887=1

2g119888119890 (120575

119905g119890119886

120575119905x119887+120575119905g

119890119887

120575119905x119886minus120575119905g

119886119887

120575119905x119890) (55)

Corollary 8 Let (1198721times119891119872

2 119865) be a twisted product Finsler

manifold Then

F119888119886119887= (F119896

119894119895 F119896

119894120573 F119896

120572119895 F119896

120572120573 F120574

119894119895 F120574

119894120573 F120574

120572119895 F120574

120572120573) (56)

6 ISRN Geometry

where

F119896119894119895= 119865

119896

119894119895minus (119872

119903

119895119862119896

119894119903+119872

119903

119894119862119896

119895119903minus119872

119903

ℎ119862119894119895119903119892119896ℎ) (57)

F119896119894120573= minusG119903

120573119862119896

119894119903= F119896

120573119894 (58)

F119896120572120573= minus119891119891

119896119892120572120573+ 119891

2119892119896ℎG120582

ℎ119862120572120573120582 (59)

F120574119894119895= 119891

minus2119892120574120582G119903

120582119862119894119895119903 (60)

F120574119894120573= 119891

minus1119891119894120575120574

120573minus G120572

119894119862120574

120572120573= F120574

120573119894 (61)

F120574120572120573= 119865

120574

120572120573+ 119873

120574

120572120573minus (119872

120583

120573119862120574

120572120583+119872

120583

120572119862120574

120573120583minus119872

120583

120582119862120572120573120583119892120574120582)

(62)

119865119896

119894119895=1

2119892119896ℎ(120575119892

ℎ119894

120575119909119895+

120575119892ℎ119895

120575119909119894minus

120575119892119894119895

120575119909ℎ)

119865120574

120572120573=1

2119892120574120582(120575119892

120582120572

120575119906120573+

120575119892120582120573

120575119906120572minus

120575119892120572120573

120575119906120582)

119872119903

119894= 119862

119903ℎ

119894119891119891

ℎ1198652

2

119872120583

120572= 119891

minus1(119862

120583120574

1205721198911205741198652

2+ 119891

119903119910119903120575120583

120572+ 119891

120574V120574120575120583

120572minus 119892

120583120574119891120574V120572+ 119891

120572V120583)

119873120574

120572120573= 119891

minus1(119891

120573120575120574

120572+ 119891

120572120575120574

120573minus 119891

120582119892120574120582119892120572120573)

(63)

Proof By using (55) we have

F119896119894119895=1

2119892119896ℎ(120575119905119892ℎ119894

120575119905119909119895+

120575119905119892ℎ119895

120575119905119909119894minus

120575119905119892119894119895

120575119905119909ℎ) (64)

Since 119892119894119895is a function with respect to (119909 119910) then by (25) and

(30) we obtain

120575119905119892ℎ119894

120575119905119909119895=120575119892

ℎ119894

120575119909119895minus 2119872

119903

119895119862ℎ119894119903 (65)

Interchanging 119894 119895 and ℎ in the previous equation gives us

120575119905119892ℎ119895

120575119905119909119894=

120575119892ℎ119895

120575119909119894minus 2119872

119903

119894119862ℎ119895119903

120575119905119892119894119895

120575119905119909ℎ=

120575119892119894119895

120575119909ℎminus 2119872

119903

ℎ119862119894119895119903

(66)

Putting these equation in (64) give us (57) In the similar waywe can prove the another relation

By using (i) of (23) and (57)ndash(62) we can conclude thefollowing

Lemma 9 Let (1198721times119891119872

2 119865) be a twisted product Finsler

manifold Then y119888F119886119887119888= G119886

119887 where F119886

119887119888and G119886

119887are defined by

(55) and (i) of (23) respectively

The Cartan torsion is one of the most important non-Riemannian quantity in Finsler geometry and it is first

introduced by Finsler and emphasized by Cartan whichmeasures a departure from a Riemannian manifold Moreprecisely a Finsler metric reduces to a Riemannian metricif and only if it has vanishing Cartan torsion The localcomponents of Cartan tensor field of the twisted Finslermanifold (119872

1times119891119872

2 119865) is defined by

C119886

119887119888=1

2g119886119890 120597g119887119890120597y119888 (67)

From this definition we conclude the following

Lemma 10 Let119862119896119894119895and119862120574

120572120573be the local components of Cartan

tensor field on1198721and119872

2 respectively Then one has

C119888

119886119887= (C119896

119894119895C119896

119894120573C119896

120572119895C119896

120572120573C120574

119894119895C120574

119894120573C120574

120572119895C120574

120572120573) (68)

where

C119896

119894119895=1

2119892119896ℎ120597119892

119894119895

120597119910ℎ= 119862

119896

119894119895

C120574

120572120573=1

2119892120574120582120597119892

120572120573

120597V120582= 119862

120574

120572120573

(69)

and C119896

119894120573= C119896

120572119895= C119896

120572120573= C120574

119894119895= C120574

119894120573= C120574

120572119895= 0

By using the Lemma 10 we can get the following

Corollary 11 Let (1198721times119891119872

2 119865) be a twisted product Finsler

manifoldThen (1198721times119891119872

2 119865) is a Riemannianmanifold if and

only if (1198721 119865

1) and (119872

2 119865

2) are Riemannian manifold

Various interesting special forms of Cartan tensors havebeen obtained by some Finslerians [11] The Finsler spaceshaving such special forms have been called C-reducible C2-like semi-C-reducible and so forth In [21] Matsumotointroduced the notion of C-reducible Finsler metrics andproved that any Randers metric is C-reducible Later onMatsumoto-Hojo proves that the converse is true too [15]

Here we define the Matsumoto twisted tensorM119886119887119888

for atwisted product Finsler manifold (119872

1times119891119872

2 119865) as follows

M119886119887119888= C

119886119887119888minus1

119899 + 1I

119886h119887119888+ I

119887h119886119888+ I

119888h119886119887 (70)

where I119886= g119887119888C

119886119887119888C

119886119887119888= g

119888119889C119889

119886119887 andh

119886119887= g

119886119887minus(1119865

2)y

119886y119887

By attention to the previous equation and relations

C119894119895119896= 119862

119894119895119896 C

120572120573120574= 119891

2119862120572120573120574 (71)

we obtain

M120572119895119896= minus

1

119899 + 1119868

120572(119892

119895119896minus1

1198652119910119895119910119896) minus1198912

1198652V120572(119868

119895119910119896+ 119868

119896119910119895)

(72)

Contracting the previous equation in 119910119895119910119896 gives us

119910119895119910119896M

120572119895119896= minus11989121198652

11198652

2

(119899 + 1) 1198652119868120572 (73)

ISRN Geometry 7

Similarly we obtain

V120582V120573M119894120573120582= minus11989121198652

11198652

2

(119899 + 1) 1198652119868119894 (74)

Therefore if M119894120573120582= M

120572119895119896= 0 then we get 119868

119894= 119868

120572= 0 that

is (1198721 119865

1) and (119872

2 119865

2) are Riemannian manifolds Thus we

have the following

Theorem 12 There is not exist any C-reducible twisted prod-uct Finsler manifold

Now we are going to consider semi-C-reducible twistedproduct Finsler manifold (119872

1times119891119872

2 119865) Let (119872

1times119891119872

2 119865) be

a semi-C-reducible twisted product Finsler manifold Thenwe have

C119886119887119888=119901

119899 + 1I

119886h119887119888+ I

119887h119886119888+ I

119888h119886119887 +119902

C2I119886I119887I119888 (75)

where C2= I119886I

119886and 119901 and 119902 are scalar function on119872

1times119891119872

2

with 119901 + 119902 = 1 This equation gives us

0 = C120572119895119896

=119901

119899 + 1119868

120572(119892

119895119896minus1

1198652119910119895119910119896) minus1198912

1198652V120572(119868

119895119910119896+ 119868

119896119910119895)

+119902

C2119868120572119868119895119868119896

(76)

Contractiing the previous equation with 119910119895119910119896 implies that

11990111989121198652

11198652

2119868120572= 0 (77)

Therefore we have 119901 = 0 or 119868120572= 0 If 119901 = 0 then

119865 is 1198622-like metric But if 119901 = 0 then 119868120572= 0 that is

1198652is Riemannian metric In this case with similar way

we conclude that 1198651is Riemannian metric But definition

119865 cannot be a Riemannian metric Therefore we have thefollowing

Theorem 13 Every semi-C-reducible twisted product Finslermanifold (119872

1times119891119872

2 119865) is a 1198622-like manifold

4 Riemannian Curvature

The Riemannian curvature of twisted product Finsler man-ifold (119872

1times119891119872

2 119865) with respect to Berwald connection is

given by

R 119886

119887 119888119889=120575119905F119886

119887119888

120575119905x119889minus120575119905F119886

119887119889

120575119905x119888+ F119886

119889119890F119890119887119888minus F119886

119888119890F119890119887119889 (78)

Lemma 14 Let (1198721times119891119872

2 119865) be a twisted product Finsler

manifold Then one has

R119886

119888119889= y119887R 119886

119887 119888119889 (79)

where R119886

119888119889and y119887R 119886

119887 119888119889are given by (50) and (78)

Proof By using (78) we have

y119887R 119894

119887 119896119897= y119887120575119905F119894

119887119896

120575119905x119897minus y119887120575119905F119894

119887119897

120575119905x119896+ y119887F119894

119897119890F119890119887119896 minus y119887F119894

119896119890F119890119887119897 (80)

By using Corollary 8 and Lemma 9 we obtain

y119887120575119905F119894

119887119896

120575119905x119897=120575119905G119894

119896

120575119905119909119897+ F119894

119895119896G119895

119897+ F119894

120573119896G120573

119897

y119887F119894119897119890F119890119887119896= F119894

119897ℎGℎ

119896+ F119894

119897120574G120574

119896

(81)

Interchanging 119894 and 119895 in the previous equation implies that

y119887120575119905F119894

119887119897

120575119905x119896=120575119905G119894

119897

120575119905119909119896+ F119894

119895119897G119895

119896+ F119894

120573119897G120573

119896

y119887F119894119896119890F119890119887119897= F119894

119896ℎGℎ

119897+ F119894

119896120574G120574

119897

(82)

Setting (81) and (82) in (80) gives us y119887R119894

119887 119896119897= R119894

119896119897 In the

similar way we can obtain this relation for another indices

Using (78) we can compute the Riemannian curvature ofa twisted product Finsler manifold

Lemma 15 Let (1198721times119891119872

2 119865) be a twisted product Finsler

manifoldThen the coefficients of Riemannian curvature are asfollows

R119894

119895 119896119897= 119877

119894

119895 119896119897

minus 119872119903

119897

120597119865119894

119895119896

120597119910119903+

120575119905119872

119894

119895119896

120575119905119909119897+119865

119894

119897ℎ119872

ℎ

119895119896+119872

119894

119897ℎ119865ℎ

119895119896minus119872

119894

119897ℎ119872

ℎ

119895119896

+ 119891minus2119892120572120574G119903

120572G119898

120574119862119894

119897119903119862119895119896119898 minusC

119896

119897

(83)

R 119894

120572 119896119897= minus

120575119905

120575119905119909119897(G119903

120572119862119894

119896119903) minus (119865

119894

119903119897minus119872

119894

119903119897)G119898

120572119862119903

119896119898

minus119891minus1G119903

120573119862119894

119897119903119891119896120575120573

120572+ G119903

120573G120583

119896119862119894

119897119903119862120573

120572120583 minus C

119896

119897

(84)

R 119894

119895 120573120582= minus

120575119905

120575119905119906120582(G119903

120573119862119894

119895119903) + G119898

120582G119897

120573119862119894

119903119898119862119903

119895119897

minus (119891119894119892120572120582minus 119891G120583

ℎ119892119894ℎ119862120572120582120583) (119891

119895120575120572

120573minus 119891G]

119895119862120572

120573])

minus C120573

120582

(85)

8 ISRN Geometry

R 119894

120572 120573119897=120575119905

120575119905119906120573(G119903

120572119862119894

119897119903) minus120575119905

120575119905119909119897119891 (119891

119894119892120572120573minus 119891G120582

ℎ119892119894ℎ119862120572120573120582)

minus G119898

120573G119904

120572119862119894

119903119898119862119903

119897119904+ (119891

119894119892120583120573minus 119891G120582

ℎ119892119894ℎ119862120583120573120582)

times (119891119897120575120583

120572minus 119891G]

119897119862120583

120572]) minus 119891119892119903ℎ(119865

119894

119903119897minus119872

119894

119903119897)

times (119891ℎ119892120572120573minus119891G120582

ℎ119862120572120573120582)minusG119903

120583119862119894

119897119903(119865

120583

120572120573+ 119873

120583

120572120573minus119872

120583

120572120573)

(86)

R 119894

119895 120573119897= minus

120575119905

120575119905119909119897(G119903

120573119862119894

119895119903) minus

120575119905

120575119905119906120573(119865

119894

119895119897minus119872

119894

119895119897)

minus (119865119894

119897119903minus119872

119894

119897119903)G119904

120573119862119903

119895119904minus 119891

minus1G119903

120572119862119894

119897119903

times (119891119895120575120572

120573minus 119891G120583

119895119862120572

120573120583) + G119904

120573119862119894

119903119904(119865

119903

119895119897minus119872

119903

119895119897)

+ 119891minus1G119903

120583119862119895119897119903(119891

119894120575120583

120573minus 119891G120582

ℎ119892119894ℎ119862120583

120573120582)

(87)

R 119894

120572 120573120582= minus

120575119905

120575119905119906120582(119891119891

119894119892120572120573minus 119891

2119892119894ℎG120583

ℎ119862120572120573120583) + 119891G119904

120582119862119894

119903119904

times(119891119903119892120572120573minus119891G120583

119897119862120572120573120583119892119903119897)minus119891 (119891

119894119892120582120583minus119891119892

119894ℎG120581

ℎ119862120582120583120581)

times (119865120583

120572120573+ 119873

120583

120572120573minus119872

120583

120572120573) minus C

120573

120582

(88)

R 120574

119895 119896119897=

120575119905

120575119905119909119897(119891

minus2119892120574120582G119903

120582119862119895119896119903) + 119891

minus2119892120574120582G119904

120582119862119897119903119904

times (119865119903

119895119896minus119872

119903

119895119896) + 119891

minus3G119903

120583119862119895119896119903(119891

119897119892120574120583minus 119891G120572

119897119862120574120583

120572)

minus C119896

119897

(89)

R 120574

119895 120573119897=120575119905

120575119905119909119897(119891

minus1119891119895120575120574

120573minus 119891G120572

119895119862120574

120572120573)

minus120575119905

120575119905119906120573(119891

minus2119892120574120582G119903

120582119862119895119897119903) minus 119891

minus2119892120574120582G119904

120582G119898

120573119862ℎ

119897119904119862ℎ119895119898

+ 119891minus2(119891

119897120575120574

120583minus 119891G120572

119897119862120574

120583120572) (119891

119895120575120583

120573minus 119891G]

119895119862120583

120573])

minus 119891minus1(119891

119903120575120574

120573minus 119891G120572

119903119862120574

120573120572) (119865

119903

119895119897minus119872

119903

119895119897)

minus 119891minus2119892120583120582G119903

120582119862119895119897119903(119865

120574

120573120583+ 119873

120574

120573120583minus119872

120574

120573120583)

(90)

R 120574

120572 120573119897=120575119905

120575119905119909119897(119865

120574

120572120573+ 119873

120574

120572120573minus119872

120574

120572120573)

minus120575119905

120575119905119906120573(119891

minus1119891119897120575120574

120572minus G120583

119897119862120574

120572120583)

minus 119891minus1119892120574120582G119904

120582119862ℎ

119897119904(119891

ℎ119892120572120573minus 119891G120583

ℎ119862120572120573120583)

+ 119891minus1(119891

119897120575120574

120583minus 119891G120581

119897119862120574

120583120581) (119865

120583

120572120573+ 119873

120583

120572120573minus119872

120583

120572120573)

+ 119891minus1G119904

120572119862119903

119897119904(119891

119903120575120574

120573minus 119891G120581

119903119862120574

120573120581)

minus 119891minus1(119891

119897120575120583

120572minus 119891G120581

119897119862120583

120572120581) (119865

120574

120573120583+ 119873

120574

120573120583minus119872

120574

120573120583)

(91)

R 120574

119895 120573120582=

120575119905

120575119905119906120582(119891

minus1119891119895120575120574

120573minus G120572

119895119862120574

120572120573)

+ 119891minus1(119865

120574

120572120582+ 119873

120574

120572120582minus119872

120574

120572120582) (119891

119895120575120572

120573minus 119891G]

119895119862120572

120573])

minus119891minus1G119898

120573119862119903

119895119898(119891

119903120575120574

120582minus 119891G120572

119903119862120574

120582120572) minus C

120573

120582

(92)

R 120574

120572 119896119897=

120575119905

120575119905119909119897(119891

minus1119891119896120575120574

120572minus G120583

119896119862120574

120572120583)

+ 119891minus2(119891

119897120575120574

120573minus 119891G120581

119897119862120574

120573120581) (119891

119896120575120573

120572minus 119891G]

119896119862120573

120572])

minus119891minus2G119904

120583G119898

120572119892120574120583119862ℎ

119897119904119862ℎ119896119898 minus C

119896

119897

(93)

R 120574

120572 120573120582= 119877

120574

120572 120573120582minus 119872

120581

120582

120597119865120574

120572120573

120597V120581+

120575119905119872

120574

120572120573

120575119905119906120582+119865

120574

120582120583119872

120583

120572120573+119872

120574

120582120583119865120583

120572120573

minus119872120574

120582120583119872

120583

120572120573+120575119905119873

120574

120572120582

120575119905119906120573+ 119865

120574

120573120583119873

120583

120572120582

+ 119873120574

120573120583119865120583

120572120582+ 119873

120574

120573120583119873

120583

120572120582+ 119873

120574

120582120583119872

120583

120572120573

+ 119873120583

120572120573119872

120574

120582120583+ (119892

119903119904120575120574

120582119891119904minus 119891119892

119903119904G120581

119904119862120574

120582120581)

times (119892120572120573119891119903minus 119891G120583

119903119862120572120573120583) minusC

120573

120582

(94)

where

119872119894

119895119896= 119872

119903

119896119862119894

119895119903+119872

119903

119895119862119894

119896119903minus119872

119903

ℎ119892119894ℎ119862119895119896119903

119872120574

120572120573= 119872

120583

120573119862120574

120572120583+119872

120583

120572119862120574

120573120583minus119872

120583

] 119892120574]119862120572120573120583

119873120574

120572120573= 119891

minus1(119891

120573120575120574

120572+ 119891

120572120575120574

120573minus 119891

120582119892120574120582119892120572120573)

(95)

and C119894

119895denotes the interchange of indices 119894 119895 and subtraction

ByTheorem 18 we have the following

Theorem 16 Let (1198721times119891119872

2 119865) be a flat twisted product

Finsler manifold and let (1198721 119865

1) be Riemannian If 119891 is a

function on1198722 only then (119872

1 119865

1) is locally flat

Similarly we get the following

Theorem 17 Let (1198721times119891119872

2 119865) be a flat twisted product

Finsler manifold and let (1198722 119865

2) be Riemannian If 119891 is a

function on 1198721 only then (119872

2 119865

2) is a space of positive

constant curvature ||119892119903119886119889119891||2

ISRN Geometry 9

Proof Since 1198722is Riemannain and 119891 is a function on 119872

1

then by (94) we obtain

R 120574

120572 120573120582= 119877

120574

120572 120573120582+1003817100381710038171003817119892119903119886119889119891

1003817100381710038171003817

2

(120575120574

120582119892120572120573minus 120575

120574

120573119892120572120582) (96)

Since (1198721times119891119872

2 119865) is flat then R120574

120572 120573120582= 0 Thus the proof is

complete

Theorem 18 Let (1198721times119891119872

2 119865) be a twisted product Rieman-

nian manifold and let 119891 be a function on 1198722 only Then

(1198721times119891119872

2 119865) is flat if and only if (119872

1 119865

1) is flat and the

Riemannian curvature of (1198722 119865

2) satisfies in the following

equation

119877120574

120572 120573120582= 120575119905119873

120574

120572120582

120575119905119906120573+ 119865

120574

120573120583119873

120583

120572120582+ 119873

120574

120573120583119865120583

120572120582+ 119873

120574

120573120583119873

120583

120572120582 minus C

120573

120582

(97)

5 Twisted Product Finsler Manifolds withNon-Riemannian Curvature Properties

There are several important non-Riemannian quantities suchas the Berwald curvature B the mean Berwald curvatureE and the Landsberg curvature L [22] They all vanishfor Riemannian metrics hence they are said to be non-Riemannian In this section we find some necessary and suf-ficient conditions underwhich a twisted product Riemannianmanifold are Berwaldian of isotropic Berwald curvatureof isotropic mean Berwald curvature First we prove thefollowing

Lemma 19 Let (1198721times119891119872

2 119865) be a twisted product Finsler

manifold Then the coefficients of Berwald curvature are asfollows

B120574

120572120573120582= 119861

120574

120572120573120582+ 119891

minus1(119862

120574]120582120572120573119891]119865

2

2+ 2119862

120574]120572120573119891]V120582

+ 2119862120574]120572120582119891]V120573 + 2119862

120574]120572119891]119892120582120573

+ 2119862120574]120582120573119891]V120572 + 2119862

120574]120573119891]119892120582120572

+ 2119862120574]120582119891]119892120572120573 minus 2119862120572120573120582119891

120574)

(98)

B119896

119894119895119897= 119861

119896

119894119895119897+ 119891119862

119896ℎ

119897119895119894119891ℎ1198652

2 (99)

B119896

119894120573119897= 2119891119862

119896ℎ

119894119897119891ℎV120573 (100)

B119896

120572120573119897= 2119891119892

120572120573119862119896ℎ

119897119891ℎ (101)

B119896

120572120573120582= minus 2119891119862

120572120573120582119891119896 (102)

B120574

119894120573120582= B120574

119894119895120582= B120574

119894119895119896= 0 (103)

Let (1198721times119891119872

2 119865) is a Berwald manifold Then we have

B119889

119886119887119888= 0 By using (102) we get

119862120572120573120582119891119896= 0 (104)

Multiplying this equation in 119892119896119903 we obtain

119862120572120573120582119891119903= 0 (105)

Thus if 119891 is not constant on1198721 then we have 119862

120572120573120582= 0 Also

from (101) we result that

119862119896ℎ

119897119891ℎ= 0 (106)

Differentiating this equation with respect to 119910119895 gives us

119862119896ℎ

119897119895119891ℎ= 0 (107)

Similarly we obtain

119862119896ℎ

119897119895119894119891ℎ= 0 (108)

Setting the last equation in (99) implies that 119861119896119894119895119897= 0 that

is (1198721 119865

1) is Berwaldian These explanations give us the

following theorem

Theorem 20 Let (1198721times119891119872

2 119865) be a twisted product Finsler

manifold and let119891 be not constant on1198721Then (119872

1times119891119872

2 119865)

is Berwaldian if and only if (1198721 119865

1) is Berwaldian (119872

2 119865

2) is

Riemannian and the equation 119862119896ℎ119897119891ℎ= 0 is hold

But if 119891 is constant on1198721 that is 119891

119894= 0 then we get the

following

Theorem 21 Let (1198721times119891119872

2 119865) be a twisted product Finsler

manifold and 119891 is constant on 1198721 Then (119872

1times119891119872

2 119865)

is Berwaldian if and only if (1198721 119865

1) is Berwaldian and

the Berwald curvature of (1198722 119865

2) satisfies in the following

equation

119861120574

120572120573120582= minus 119891

minus1(119862

120574]120573120572120582119891]119865

2

2+ 2119862

120574]120573120572119891]V120582 + 2119862

120574]120582120572119891]V120573

+ 2119862120574]120572119891]119892120582120573 + 2119862

120574]120573120582119891]V120572

+ 2119862120574]120573119891]119892120582120572 + 2119862

120574]120582119891]119892120572120573

minus 2119892120574]119862120572120573120582119891])

(109)

Here we consider twisted product Finsler manifold(119872

1times119891119872

2 119865) of isotropic Berwald curvature

Theorem 22 Every isotropic Berwald twisted product Finslermanifold (119872

1times119891119872

2 119865) is a Berwald manifold

Proof Let (1198721times119891119872

2 119865) be an isotropic Berwald manifold

Then we have

B119889

119886119887119888= 119888119865

minus1h119889

119886h119887119888+ h119889

119887h119886119888+ h119889

119888h119886119887+ 2C

119886119887119888y119889 (110)

where 119888 = 119888(x) is a function on119872 Setting 119886 = 119895 119887 = 119896 119888 = 119897and 119889 = 120574 and using (103) imply that

119888119865minus13

1198652119910119895119910119896119910119897V120574 minus V120574 (119910

119895119892119896119897+ 119910

119896119892119895119897+ 119910

119897119892119895119896) = 0 (111)

Multiplying the previous equation in 119910119895119910119896 we derive that119888119891

21198652

11198652

2= 0 Thus we have 119888 = 0 that is (119872

1times119891119872

2) is

Berwaldian

10 ISRN Geometry

Now we are going to study twisted product Finslermanifold of isotropicmean Berwald curvature For this workwe must compute the coefficients of mean Berwald curvatureof a twisted product Finsler manifold

Lemma 23 Let (1198721times119891119872

2 119865) be a twisted product Finsler

manifold Then the coefficients of mean Berwald curvature areas follows

E120572120573= 119864

120572120573+ 119891119892

120572120573119868ℎ119891ℎ+1

2119891119868

]120572120573119891]119865

2

2+ 119891

minus1119891]

times (119862120574]120572120573

V120574+ 119868

]120572V120573+ 119868

]120573V120572+ 119862

]120572120573+ 119868

]119892120572120573)

(112)

E119894119895= 119864

119894119895+1

2119891119868

ℎ

119895119894119891ℎ1198652

2 (113)

E119894120573= 119891119868

ℎ

119894119891ℎV120573 (114)

where 119864119894119895

and 119864120572120573

are the coefficients of mean Berwaldcurvature of (119872

1 119865

1) and (119872

2 119865

2) respectively

Proof By definition and Lemma 19 we get the proof

Theorem24 The twisted product Finslermanifold (1198721times119891119872

2

119865) is weakly Berwald if and only if (1198721 119865

1) is weakly Berwald

119868ℎ119891ℎ= 0 and the following hold

119864120572120573= minus

1

2119891119868

]120572120573119891]119865

2

2minus 119891

minus1119891]

times (119862120574]120572120573

V120574+ 119868

]120572V120573+ 119868

]120573V120572+ 119862

]120572120573+ 119868

]119892120572120573)

(115)

Proof If (1198721times119891119872

2) be a weakly Berwald manifold then we

have

E120572120573= E

119894119895= E

119894120573= 0 (116)

Thus by using (114) we result that 119868ℎ119894119891ℎ= 0 This equation

implies that

119868ℎ

119895119894119891ℎ= 0 119868

ℎ119891ℎ= 0 (117)

By setting these equations in (112) and (113) we conclude that119864119894119895= 0 and 119864

120572120573satisfies in (115)

Now if 119891 is constant on1198722 then (115) implies that 119864

120572120573=

0 Thus we conclude the following

Corollary 25 Let (1198721times119891119872

2 119865) be a twisted product Finsler

manifold and let 119891 be a function on 1198721 only Then

(1198721times119891119872

2 119865) is weakly Berwald if and only if (119872

1 119865

1) and

(1198722 119865

2) are weakly Berwald manifolds and 119868ℎ119891

ℎ= 0

Now we consider twisted product Finsler manifolds withisotropic mean Berwald curvature It is remarkable that as aconsequence of Lemma 23 we have the following

Lemma26 Twisted product Finslermanifold (1198721times119891119872

2 119865) is

isotropic mean Berwald manifold if and only if

119864120572120573+ 119891119892

120572120573119868ℎ119891ℎ+1

2119891119868

]120572120573119891]119865

2

2+ 119891

minus1119891]

times (119862120574]120572120573

V120574+ 119868

]120572V120573+ 119868

]120573V120572+ 119862

]120572120573+ 119868

]119892120572120573)

minus119899 + 1

2119888119891

2119865minus1(119892

120572120573minus1198912

1198652V120572V120573) = 0

(118)

119864119894119895+1

2119891119868

ℎ

119895119894119891ℎ1198652

2minus119899 + 1

2119888119865

minus1(119892

119894119895minus1

1198652119910119894119910119895) = 0 (119)

119888 (119899 + 1) 119865minus3119910119894+ 119891119868

ℎ

119894119891ℎ= 0 (120)

where 119888 = 119888(x) is a scalar function on119872

Theorem 27 Every twisted product Finsler manifold(119872

1times119891119872

2 119865) with isotropic mean Berwald curvature is a

weakly Berwald manifold

Proof Suppose that 119865 is isotropic mean Berwald twistedproduct Finsler metric Then differentiating (120) withrespect to V120574 gives us

119888 (119899 + 1) 1198912119865minus5V

120574119910119894= 0 (121)

Thus we conclude that 119888 = 0 This implies that 119865 reduces to aweakly Berwald metric

6 Locally Dually Flat Twisted ProductFinsler Manifolds

In [23] Amari and Nagaoka introduced the notion of duallyflat Riemannian metrics when they study the informationgeometry on Riemannian manifolds Information geometryhas emerged from investigating the geometrical structure ofa family of probability distributions and has been appliedsuccessfully to various areas including statistical inferencecontrol system theory andmultiterminal information theoryIn Finsler geometry Shen extends the notion of locally duallyflatness for Finsler metrics [17] Dually flat Finsler metricsform a special and valuable class of Finsler metrics in Finslerinformation geometry which play a very important role instudying flat Finsler information structure [24 25]

In this section we study locally dually flat twisted productFinsler metrics It is remarkable that a Finsler metric 119865 =119865(x y) on a manifold119872 is said to be locally dually flat if atany point there is a standard coordinate system (x119886 y119886) in119879119872such that it satisfies

12059721198652

120597x119887120597y119886y119887 = 2120597119865

2

120597x119886 (122)

In this case the coordinate (x119886) is called an adapted localcoordinate system Byusing (122) we can obtain the followinglemma

ISRN Geometry 11

Lemma 28 Let (1198721times119891119872

2 119865) be a twisted product Finsler

manifold Then 119865 is locally dually flat if and only if 1198651and 119865

2

satisfy in the following equations

12059721198652

1

120597119909119896120597119910119897119910119896= 2120597119865

2

1

120597119909119897+ 4119891119891

1198971198652

2 (123)

4119891119896V120573119910119896+ 119891

12059721198652

2

120597119906120572120597V120573V120572 + 4119891

120572V120573V120572 = 2119891

1205971198652

2

120597119906120573+ 4119891

1205731198652

2

(124)

Now let 119865 be a locally dually flat Finsler metric Takingderivative with respect to V120574 from (123) yields 119891

119897= 0 which

means that 119891 is a constant function on1198721 In this case the

relations (123) and (124) reduce to the following

12059721198652

1

120597119909119896120597119910119897119910119896= 2120597119865

2

1

120597119909119897 (125)

11989112059721198652

2

120597119906120572120597V120573V120572 + 4119891

120572V120573V120572 = 2119891

1205971198652

2

120597119906120573+ 4119891

1205731198652

2 (126)

By (125) we deduce that 1198651is locally dually flat

Now we assume that 1198651and 119865

2are locally dually flat

Finsler metrics Then we have

12059721198652

1

120597119909119896120597119910119897119910119896= 2120597119865

2

1

120597119909119897

12059721198652

2

120597119906120572120597V120573V120572 = 2

1205971198652

2

120597119906120573

(127)

By (127) we derive that (123) and (124) are hold if and only ifthe following hold

119891119897= 0 119891

120572V120573V120572 = 119891

1205731198652

2 (128)

Therefore we can conclude the following

Theorem 29 Let (1198721times119891119872

2 119865) be a twisted product Finsler

manifold

(i) If 119865 is locally dually flat then 1198651is locally dually flat 119891

is a function with respect to (119906120572) only and 1198652satisfies

in (126)(ii) If 119865

1and 119865

2are locally dually flat then 119865 is locally

dually flat if and only if 119891 is a function with respect(119906

120572) only and 119865

2satisfies in (128)

ByTheorem 29 we conclude the following

Corollary 30 There is not exist any locally dually flat propertwisted product Finsler manifold

References

[1] J K Beem P E Ehrlich and K L Easley Global LorentzianGeometry vol 202 of Monographs and Textbooks in Pure andApplied Mathematics Marcel Dekker New York NY USA 2ndedition 1996

[2] G Ganchev and V Mihova ldquoRiemannian manifolds of quasi-constant sectional curvaturesrdquo Journal fur die Reine und Ange-wandte Mathematik vol 522 pp 119ndash141 2000

[3] B OrsquoNeill Semi-Riemannian Geometry With Applications toRelativity vol 103 of Pure and Applied Mathematics AcademicPress New York NY USA 1983

[4] M Lohnherr and H Reckziegel ldquoOn ruled real hypersurfacesin complex space formsrdquo Geometriae Dedicata vol 74 no 3pp 267ndash286 1999

[5] B-Y Chen F Dillen L Verstraelen and L VranckenldquoLagrangian isometric immersions of a real-space-form119872119899

(119888)

into a complex-space-form119872119899(4119888)rdquoMathematical Proceedings

of the Cambridge Philosophical Society vol 124 no 1 pp 107ndash125 1998

[6] N Koike ldquoThe decomposition of curvature netted hypersur-facesrdquo Geometriae Dedicata vol 54 no 1 pp 1ndash11 1995

[7] B-y Chen Geometry of Submanifolds and its ApplicationsScience University of Tokyo Tokyo Japan 1981

[8] R Ponge and H Reckziegel ldquoTwisted products in pseudo-Riemannian geometryrdquo Geometriae Dedicata vol 48 no 1 pp15ndash25 1993

[9] B-Y Chen ldquoTwisted product CR-submanifolds in Kaehlermanifoldsrdquo Tamsui Oxford Journal of Mathematical Sciencesvol 16 no 2 pp 105ndash121 2000

[10] L Kozma I R Peter and H Shimada ldquoOn the twisted productof Finsler manifoldsrdquo Reports on Mathematical Physics vol 57no 3 pp 375ndash383 2006

[11] M Matsumoto and C Shibata ldquoOn semi-C-reducibility T-tensor= 0 and S4-likeness of Finsler spacesrdquo Journal of Math-ematics of Kyoto University vol 19 no 2 pp 301ndash314 1979

[12] B Najafi Z Shen and A Tayebi ldquoFinsler metrics of scalar flagcurvature with special non-Riemannian curvature propertiesrdquoGeometriae Dedicata vol 131 pp 87ndash97 2008

[13] E Peyghan andAHeydari ldquoConformal vector fields on tangentbundle of a Riemannian manifoldrdquo Journal of MathematicalAnalysis and Applications vol 347 no 1 pp 136ndash142 2008

[14] Z Shen Differential Geometry of Spray and Finsler SpacesKluwer Academic Publishers Dordrecht The Netherlands2001

[15] M Matsumoto and S-I Hojo ldquoA conclusive theorem on C-reducible Finsler spacesrdquo The Tensor Society vol 32 no 2 pp225ndash230 1978

[16] D Bao S-S Chern and Z Shen An Introduction to Riemann-Finsler Geometry vol 200 of Graduate Texts in MathematicsSpringer New York NY USA 2000

[17] Z Shen ldquoRiemann-Finsler geometry with applications to infor-mation geometryrdquo Chinese Annals of Mathematics B vol 27 no1 pp 73ndash94 2006

[18] Y Ichijyo ldquoFinsler manifolds modeled on a Minkowski spacerdquoJournal of Mathematics of Kyoto University vol 16 no 3 pp639ndash652 1976

[19] X Chen and Z Shen ldquoOn Douglas metricsrdquo PublicationesMathematicae Debrecen vol 66 no 3-4 pp 503ndash512 2005

[20] L Berwald ldquoUber Parallelubertragung in Raumen mit allge-meiner Massbestimmungrdquo Jahresbericht der DMVmdashDeutscheMathematiker-Vereinigung vol 34 pp 213ndash220 1926

[21] M Matsumoto ldquoOn Finsler spaces with Randersrsquo metric andspecial forms of important tensorsrdquo Journal of Mathematics ofKyoto University vol 14 pp 477ndash498 1974

12 ISRN Geometry

[22] A Tayebi and E Peyghan ldquoOn special Berwald metricsrdquoSymmetry Integrability and Geometry vol 6 article 008 9pages 2010

[23] S-I Amari and H NagaokaMethods of Information Geometryvol 191 of Translations of Mathematical Monographs AmericanMathematical Society Providence RI USA 2000

[24] X Cheng Z Shen and Y Zhou ldquoOn locally dually flat Finslermetricsrdquo International Journal ofMathematics vol 21 no 11 pp1531ndash1543 2010

[25] Q Xia ldquoOn a class of locally dually flat Finsler metrics ofisotropic flag curvaturerdquo Publicationes Mathematicae Debrecenvol 78 no 1 pp 169ndash190 2011

Submit your manuscripts athttpwwwhindawicom

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Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

ISRN Geometry 3

A Finsler metric 119865 is said to be isotropic Berwald metricand isotropic mean Berwald metric if its Berwald curvatureand mean Berwald curvature are in the following formrespectively

119861119894

119895119896119897= 119888 119865

119910119895119910119896120575

119894

119897+ 119865

119910119896119910119897120575

119894

119895+ 119865

119910119897119910119895120575

119894

119896+ 119865

119910119895119910119896119910119897119910

119894

119864119894119895=1

2(119899 + 1) 119888119865

minus1ℎ119894119895

(10)

where 119888 = 119888(119909) is a scalar function on119872 [19]The Riemann curvature R

119910= 119877

119894

119896119889119909

119896otimes (120597120597119909

119894)|119909

119879119909119872 rarr 119879

119909119872 is a family of linear maps on tangent spaces

defined by

119877119894

119896= 2120597119866

119894

120597119909119896minus 119910

119895 1205972119866119894

120597119909119895120597119910119896+ 2119866

119895 1205972119866119894

120597119910119895120597119910119896

minus120597119866

119894

120597119910119895

120597119866119895

120597119910119896

(11)

The flag curvature in Finsler geometry is a natural extensionof the sectional curvature in Riemannian geometry was firstintroduced by L Berwald [20] For a flag 119875 = span119910 119906 sub119879119909119872with flagpole119910 the flag curvatureK = K(119875 119910) is defined

by

K (119875 119910) =g119910(119906R

119910(119906))

g119910(119910 119910) g

119910(119906 119906) minus g

119910(119910 119906)

2 (12)

We say that a Finsler metric 119865 is of scalar curvature if for any119910 isin 119879

119909119872 the flag curvature K = K(119909 119910) is a scalar function

on the slit tangent bundle119879119872∘ IfK = constant then119865 is saidto be of constant flag curvature

3 Nonlinear Connection

Let (1198721 119865

1) and (119872

2 119865

2) be two Finsler manifolds Then the

functions

(i) 119892119894119895(119909 119910) =

1

2

12059721198652

1(119909 119910)

120597119910119894120597119910119895

(ii) 119892120572120573(119906 V) =

1

2

12059721198652

2(119906 V)

120597V120572120597V120573

(13)

define a Finsler tensor field of type (0 2) on 119879119872∘

1and

119879119872∘

2 respectively Now let (119872

1times119891119872

2 119865) be a doubly warped

Finsler manifold x = (119909 119906) isin 119872 y = (119910 V) isin 119879x119872119872 = 119872

1times 119872

2 and 119879x119872 = 1198791199091198721

oplus 119879119906119872

2 Then by using

(13) we conclude that

(g119886119887(119909 119906 119910 V)) = (

1

2

12059721198652(119909 119906 119910 V)

120597y119886y119887) = [

119892119894119895

0

0 1198912119892120572120573

]

(14)

where y119886 = (119910119894 V120572) g119894119895= 119892

119894119895 g

120572120573= 119891

2119892120572120573 g

119894120573= g

120572119895=

0 119894 119895 isin 1 1198991 120572 120573 isin 1 119899

2 and 119886 119887 isin

1 1198991+ 119899

2

Now we consider spray coefficients of 1198651 119865

2 and 119865 as

119866119894(119909 119910) =

1

4119892119894ℎ(12059721198652

1

120597119910ℎ120597119909119895119910119895minus120597119865

2

1

120597119909ℎ) (119909 119910) (15)

119866120572(119906 V) =

1

4119892120572120574(12059721198652

2

120597V120574120597119906120573V120573 minus

1205971198652

2

120597119906120574) (119906 V) (16)

G119886(x y) = 1

4g119886119887 ( 120597

21198652

120597y119887120597x119888y119888 minus 120597119865

2

120597x119887) (x y) (17)

Taking into account the homogeneity of both 11986521and 1198652

2

and using (15) and (16) we can conclude that 119866119894 and 119866120572are positively homogeneous of degree two with respect to(119910

119894) and (V120572) respectively Hence from Euler theorem for

homogeneous functions we infer that

120597119866119894

120597119910119895119910119895= 2119866

119894

120597119866120572

120597V120573V120573 = 2119866120572 (18)

By setting 119886 = 119894 in (17) we have

G119894(119909 119906 119910 V) =

1

4g119894ℎ ( 120597

21198652

120597119910ℎ120597119909119895119910119895+12059721198652

120597119910ℎ120597119906120572V120572 minus

1205971198652

120597119909ℎ)

(19)Direct calculations give us

1205971198652

120597119909ℎ=120597119865

2

1

120597119909ℎ+120597119891

2

120597119909ℎ1198652

2

12059721198652

120597119910ℎ120597119909119895=12059721198652

1

120597119910ℎ120597119909119895

12059721198652

120597119910ℎ120597119906120572= 0

(20)

Putting these equations together g119894ℎ = 119892119894ℎ in the previousequation and using (15) imply that

G119894(119909 119906 119910 V) = 119866119894 (119909 119910) minus

1

2119891119891

1198941198652

2 (21)

Similarly by setting 119886 = 120572 in (17) and using (16) we obtainG120572(119909 119906 119910 V) = 119866120572 (119906 V)

+ 119891minus1(119891

119895V120572119910119895 + 119891

120582V120572V120582 minus

1

21198911205741198921205721205741198652

2)

(22)

where 119891119894= 120597119891120597119909

119894 119891120574= 120597119891120597119906

120574 119891119894 = 119892119894ℎ119891ℎ and 119891120574 = 119892120582120574119891

120582

Therefore we have G119886= (G119894

G120572) where G119886 G119894 and G120572 are

given by (17) (21) and (22) respectivelyNow we put

(i) G119886

119887=120597G119886

120597y119887

(ii) 119866119894119895=120597119866

119894

120597119910119895

(iii) 119866120572120573=120597119866

120572

120597V120573

(23)

Then we have the following

4 ISRN Geometry

Lemma 1 The coefficients G119886

119887defined by (23) satisfy in the

following

(G119886

119887(119909 119906 119910 V)) = [

G119894

119895(119909 119906 119910 V) G120572

119895(119909 119906 119910 V)

G119894

120573(119909 119906 119910 V) G120572

120573(119909 119906 119910 V)] (24)

where

G119894

119895(119909 119906 119910 V) =

120597G119894

120597119910119895= 119866

119894

119895+ 119862

119894ℎ

119895119891119891

ℎ1198652

2 (25)

G119894

120573(119909 119906 119910 V) =

120597G119894

120597V120573= minus119891119891

119894V120573 (26)

G120572

119895(119909 119906 119910 V) =

120597G120572

120597119910119895= 119891

minus1119891119895V120572 (27)

G120572

120573(119909 119906 119910 V) =

120597G120572

120597V120573

= 119866120572

120573+ 119891

minus1(119862

120572120574

1205731198911205741198652

2+ 119891

119895119910119895120575120572

120573

minus 119891120572V

120573+ 119891

120573V120572 + 119891

120574V120574120575120572

120573)

(28)

Next 119881119879119872∘ kernel of the differential of the projectionmap

120587 = (1205871 120587

2) 119879119872

∘

1oplus 119879119872

∘

2997888rarr 119872

1times119872

2 (29)

which is a well-defined subbundle of 119879119879119872∘ is consid-ered Locally Γ(119881119879119872∘

) is spanned by the natural vectorfields 1205971205971199101 1205971205971199101198991 120597120597V1 120597120597V1198992 and it is calledthe twisted vertical distribution on 119879119872∘ Then using thefunctions given by (25)ndash(28) the nonholonomic vector fieldsare defined as follows

120575119905

120575119905119909119894=120597

120597119909119894minus G119895

119894

120597

120597119910119895minus G120573

119894

120597

120597V120573 (30)

120575119905

120575119905119906120572=120597

120597119906120572minus G119895

120572

120597

120597119910119895minus G120573

120572

120597

120597V120573 (31)

which make it possible to construct a complementary vectorsubbundle119867119879119872∘ to 119881119879119872∘ in 119879119879119872∘ as follows

119867119879119872∘= span 120575

119905

1205751199051199091

120575119905

1205751199051199091198991120575119905

1205751199051199061

120575119905

1205751199051199061198992 (32)

119867119879119872∘ is called the twisted horizontal distribution on 119879119872∘

Thus the tangent bundle of 119879119872∘ admits the decomposition

119879119879119872∘= 119867119879119872

∘oplus 119881119879119872

∘ (33)

It is shown thatG = (G119886

119887) is a nonlinear connection on119879119872 =

1198791198721oplus 119879119872

2 In the following we compute the nonlinear

connection of a twisted product Finsler manifold

Proposition 2 If (1198721times119891119872

2 119865) is a twisted product Finsler

manifold then G = (G119886

119887) is the nonlinear connection on 119879119872

Further one has

120597G119894

119895

120597119910119896119910119896+

120597G119894

119895

120597V120574V120574 = G119894

119895

120597G119894

120573

120597119910119896119910119896+

120597G119894

120573

120597V120574V120574 = G119894

120573

120597G120572

119895

120597119910119896119910119896+

120597G120572

119895

120597V120574V120574 = G120572

119895

120597G120572

120573

120597119910119896119910119896+

120597G120572

120573

120597V120574V120574 = G120572

120573

(34)

Definition 3 Using decomposition (33) the twisted verticalmorphism V119905 119879119879119872∘

rarr 119881119879119872∘ is defined by

V119905 =120597

120597119910119894otimes 120575

119905119910119894+120597

120597V120572otimes 120575

119905V120572 (35)

where

120575119905119910119894= 119889119910

119894+ G119894

119895119889119909

119895+ G119894

120573119889119906

120573

120575119905V120572 = 119889V120572 + G120572

119895119889119909

119895+ G120572

120573119889119906

120573

(36)

For this projective morphism the following hold

V119905 (120597

120597119910119894) =

120597

120597119910119894 V119905 (

120597

120597V120572) =

120597

120597V120572

V119905 (120575119905

120575119905119909119894) = 0 V119905 (

120575119905

120575119905119906119894) = 0

(37)

From the previous equations we conclude that

(V119905)2

= V119905 ker (V119905) = 119867119879119872∘ (38)

This mapping is called the twisted vertical projective

Definition 4 Using decomposition (33) the doubly warpedhorizontal projective ℎ119905 119879119879119872∘

rarr 119867119879119872∘ is defined by

ℎ119905= 119894119889 minus V119905 (39)

or

ℎ119905=120575119905

120575119905119909119894otimes 119889119909

119894+120575119905

120575119905119906120572otimes 119889119906

120572 (40)

For this projective morphism the following hold

ℎ119905(120575119905

120575119905119909119894) =

120575119905

120575119905119909119894 ℎ

119905(120575119905

120575119905119906120572) =

120575119905

120575119905119906120572

ℎ119905(120597

120597119910119894) = 0 ℎ

119905(120597

120597V120572) = 0

(41)

Thus we result that

(ℎ119905)2

= ℎ119905 ker (ℎ119905) = 119881119879119872∘

(42)

ISRN Geometry 5

Definition 5 Using decomposition (33) the twisted almosttangent structure 119869119905 119867119879119872∘

rarr 119881119879119872∘ is defined by

119869119905120597

120597119910119894otimes 119889119909

119894+120597

120597V120572otimes 119889119906

120572 (43)

or

119869119905(120575119905

120575119905119909119894) =

120597

120597119910119894 119869

119905(120575119905

120575119905119906120572) =

120597

120597V120572

119869119905(120597

120597119910119894) = 119869

119905(120597

120597V120572) = 0

(44)

Thus we result that

(119869119905)2

= 0 ker 119869119905 = 119868119898119869119905 = 119881119879119872∘ (45)

Here we introduce some geometrical objects of twistedproduct Finsler manifold In order to simplify the equationswe rewritten the basis of119867119879119872∘ and 119881119879119872∘ as follows

120575119905

120575119905x119886=120575119905

120575119905119909119894120575119894

119886+120575119905

120575119905119906120572120575120572

119886

120597

120597y119886=120597

120597119910119894120575119894

119886+120597

120597V120572120575120572

119886

(46)

Thus

119879119879119872∘= span 120575

119905

120575119905x119886120597

120597y119886 (47)

The Lie brackets of this basis is given by

[120575119905

120575119905x119886120575119905

120575119905x119887] = R119888

119886119887

120597

120597y119888

[120575119905

120575119905x119886120597

120597y119887] = G119888

119886119887

120597

120597y119888

[120597

120597y119886120597

120597y119887] = 0

(48)

where

(i) R119888

119886119887=120575119905G119888

119886

120575119905x119887minus120575119905G119888

119887

120575119905x119886 (49)

(ii) G119888

119886119887=120597G119888

119886

120597y119887 (50)

Therefore we have the following

Corollary 6 Let (1198721times119891119872

2 119865) be a twisted product Finsler

manifold Then

R119888

119886119887= (R119896

119894119895R119896

119894120573R119896

120572119895R119896

120572120573R120574

119894119895R120574

119894120573R120574

120572119895R120574

120572120573)

(51)

where

R119896

119894119895=120575119905G119896

119894

120575119905119909119895minus

120575119905G119896

119895

120575119905119909119894 R119896

119894120573=120575119905G119896

119894

120575119905119906120573minus

120575119905G119896

120573

120575119905119909119894

R119896

120572119895=120575119905G119896

120572

120575119905119909119895minus

120575119905G119896

119895

120575119905119906120572 R119896

120572120573=120575119905G119896

120572

120575119905119906120573minus

120575119905G119896

120573

120575119905119906120572

R120574

119894119895=120575119905G120574

119894

120575119905119909119895minus

120575119905G120574

119895

120575119905119909119894 R120574

119894120573=120575119905G120574

119894

120575119905119906120573minus

120575119905G120574

120573

120575119905119909119894

R120574

120572119895=120575119905G120574

120572

120575119905119909119895minus

120575119905G120574

119895

120575119905119906120572 R120574

120572120573=120575119905G120574

120572

120575119905119906120573minus

120575119905G120574

120573

120575119905119906120572

(52)

With a simple calculation we have the following

Corollary 7 Let (1198721times119891119872

2 119865) be a twisted product Finsler

manifold Then

G119888

119886119887= (G119896

119894119895G119896

119894120573G119896

120572119895G119896

120572120573G120574

119894119895G120574

119894120573G120574

120572119895G120574

120572120573) (53)

where

G120574

120572120573=120597G120574

120572

120597V120573

= 119866120574

120572120573+ 119891

minus1(119862

120574120582

1205721205731198911205821198652

2+ 2119862

120574120582

120572119891120582V120573+ 2119862

120574120582

120573119891120582V120572

minus 119891120574119892120572120573+ 119891

120573120575120574

120572+ 119891

120572120575120574

120573) = G120574

120573120572

G119896

119894119895=120597G119896

119894

120597119910119895= 119866

119896

119894119895+ 119862

119896ℎ

119894119895119891119891

ℎ1198652

2= G119896

119895119894

G119896

119894120573=120597G119896

119894

120597V120573= 2119862

119896ℎ

119894119891119891

ℎV120573= G119896

120573119894

G119896

120572120573=120597G119896

120572

120597V120573= minus119891119891

119896119892120572120573= G119896

120573120572

G120574

119894120573=120597G120574

119894

120597V120573= 119891

minus1119891119894120575120574

120573= G120574

120573119894

G120574

119894119895=120597G120574

119894

120597119910119895= G120574

119895119894= 0

(54)

where 119862119896ℎ119894119895= 120597119862

119896ℎ

119894120597119910

119895 Apart from G119888

119886119887 the functions F119888

119886119887are

given by

F119888119886119887=1

2g119888119890 (120575

119905g119890119886

120575119905x119887+120575119905g

119890119887

120575119905x119886minus120575119905g

119886119887

120575119905x119890) (55)

Corollary 8 Let (1198721times119891119872

2 119865) be a twisted product Finsler

manifold Then

F119888119886119887= (F119896

119894119895 F119896

119894120573 F119896

120572119895 F119896

120572120573 F120574

119894119895 F120574

119894120573 F120574

120572119895 F120574

120572120573) (56)

6 ISRN Geometry

where

F119896119894119895= 119865

119896

119894119895minus (119872

119903

119895119862119896

119894119903+119872

119903

119894119862119896

119895119903minus119872

119903

ℎ119862119894119895119903119892119896ℎ) (57)

F119896119894120573= minusG119903

120573119862119896

119894119903= F119896

120573119894 (58)

F119896120572120573= minus119891119891

119896119892120572120573+ 119891

2119892119896ℎG120582

ℎ119862120572120573120582 (59)

F120574119894119895= 119891

minus2119892120574120582G119903

120582119862119894119895119903 (60)

F120574119894120573= 119891

minus1119891119894120575120574

120573minus G120572

119894119862120574

120572120573= F120574

120573119894 (61)

F120574120572120573= 119865

120574

120572120573+ 119873

120574

120572120573minus (119872

120583

120573119862120574

120572120583+119872

120583

120572119862120574

120573120583minus119872

120583

120582119862120572120573120583119892120574120582)

(62)

119865119896

119894119895=1

2119892119896ℎ(120575119892

ℎ119894

120575119909119895+

120575119892ℎ119895

120575119909119894minus

120575119892119894119895

120575119909ℎ)

119865120574

120572120573=1

2119892120574120582(120575119892

120582120572

120575119906120573+

120575119892120582120573

120575119906120572minus

120575119892120572120573

120575119906120582)

119872119903

119894= 119862

119903ℎ

119894119891119891

ℎ1198652

2

119872120583

120572= 119891

minus1(119862

120583120574

1205721198911205741198652

2+ 119891

119903119910119903120575120583

120572+ 119891

120574V120574120575120583

120572minus 119892

120583120574119891120574V120572+ 119891

120572V120583)

119873120574

120572120573= 119891

minus1(119891

120573120575120574

120572+ 119891

120572120575120574

120573minus 119891

120582119892120574120582119892120572120573)

(63)

Proof By using (55) we have

F119896119894119895=1

2119892119896ℎ(120575119905119892ℎ119894

120575119905119909119895+

120575119905119892ℎ119895

120575119905119909119894minus

120575119905119892119894119895

120575119905119909ℎ) (64)

Since 119892119894119895is a function with respect to (119909 119910) then by (25) and

(30) we obtain

120575119905119892ℎ119894

120575119905119909119895=120575119892

ℎ119894

120575119909119895minus 2119872

119903

119895119862ℎ119894119903 (65)

Interchanging 119894 119895 and ℎ in the previous equation gives us

120575119905119892ℎ119895

120575119905119909119894=

120575119892ℎ119895

120575119909119894minus 2119872

119903

119894119862ℎ119895119903

120575119905119892119894119895

120575119905119909ℎ=

120575119892119894119895

120575119909ℎminus 2119872

119903

ℎ119862119894119895119903

(66)

Putting these equation in (64) give us (57) In the similar waywe can prove the another relation

By using (i) of (23) and (57)ndash(62) we can conclude thefollowing

Lemma 9 Let (1198721times119891119872

2 119865) be a twisted product Finsler

manifold Then y119888F119886119887119888= G119886

119887 where F119886

119887119888and G119886

119887are defined by

(55) and (i) of (23) respectively

The Cartan torsion is one of the most important non-Riemannian quantity in Finsler geometry and it is first

introduced by Finsler and emphasized by Cartan whichmeasures a departure from a Riemannian manifold Moreprecisely a Finsler metric reduces to a Riemannian metricif and only if it has vanishing Cartan torsion The localcomponents of Cartan tensor field of the twisted Finslermanifold (119872

1times119891119872

2 119865) is defined by

C119886

119887119888=1

2g119886119890 120597g119887119890120597y119888 (67)

From this definition we conclude the following

Lemma 10 Let119862119896119894119895and119862120574

120572120573be the local components of Cartan

tensor field on1198721and119872

2 respectively Then one has

C119888

119886119887= (C119896

119894119895C119896

119894120573C119896

120572119895C119896

120572120573C120574

119894119895C120574

119894120573C120574

120572119895C120574

120572120573) (68)

where

C119896

119894119895=1

2119892119896ℎ120597119892

119894119895

120597119910ℎ= 119862

119896

119894119895

C120574

120572120573=1

2119892120574120582120597119892

120572120573

120597V120582= 119862

120574

120572120573

(69)

and C119896

119894120573= C119896

120572119895= C119896

120572120573= C120574

119894119895= C120574

119894120573= C120574

120572119895= 0

By using the Lemma 10 we can get the following

Corollary 11 Let (1198721times119891119872

2 119865) be a twisted product Finsler

manifoldThen (1198721times119891119872

2 119865) is a Riemannianmanifold if and

only if (1198721 119865

1) and (119872

2 119865

2) are Riemannian manifold

Various interesting special forms of Cartan tensors havebeen obtained by some Finslerians [11] The Finsler spaceshaving such special forms have been called C-reducible C2-like semi-C-reducible and so forth In [21] Matsumotointroduced the notion of C-reducible Finsler metrics andproved that any Randers metric is C-reducible Later onMatsumoto-Hojo proves that the converse is true too [15]

Here we define the Matsumoto twisted tensorM119886119887119888

for atwisted product Finsler manifold (119872

1times119891119872

2 119865) as follows

M119886119887119888= C

119886119887119888minus1

119899 + 1I

119886h119887119888+ I

119887h119886119888+ I

119888h119886119887 (70)

where I119886= g119887119888C

119886119887119888C

119886119887119888= g

119888119889C119889

119886119887 andh

119886119887= g

119886119887minus(1119865

2)y

119886y119887

By attention to the previous equation and relations

C119894119895119896= 119862

119894119895119896 C

120572120573120574= 119891

2119862120572120573120574 (71)

we obtain

M120572119895119896= minus

1

119899 + 1119868

120572(119892

119895119896minus1

1198652119910119895119910119896) minus1198912

1198652V120572(119868

119895119910119896+ 119868

119896119910119895)

(72)

Contracting the previous equation in 119910119895119910119896 gives us

119910119895119910119896M

120572119895119896= minus11989121198652

11198652

2

(119899 + 1) 1198652119868120572 (73)

ISRN Geometry 7

Similarly we obtain

V120582V120573M119894120573120582= minus11989121198652

11198652

2

(119899 + 1) 1198652119868119894 (74)

Therefore if M119894120573120582= M

120572119895119896= 0 then we get 119868

119894= 119868

120572= 0 that

is (1198721 119865

1) and (119872

2 119865

2) are Riemannian manifolds Thus we

have the following

Theorem 12 There is not exist any C-reducible twisted prod-uct Finsler manifold

Now we are going to consider semi-C-reducible twistedproduct Finsler manifold (119872

1times119891119872

2 119865) Let (119872

1times119891119872

2 119865) be

a semi-C-reducible twisted product Finsler manifold Thenwe have

C119886119887119888=119901

119899 + 1I

119886h119887119888+ I

119887h119886119888+ I

119888h119886119887 +119902

C2I119886I119887I119888 (75)

where C2= I119886I

119886and 119901 and 119902 are scalar function on119872

1times119891119872

2

with 119901 + 119902 = 1 This equation gives us

0 = C120572119895119896

=119901

119899 + 1119868

120572(119892

119895119896minus1

1198652119910119895119910119896) minus1198912

1198652V120572(119868

119895119910119896+ 119868

119896119910119895)

+119902

C2119868120572119868119895119868119896

(76)

Contractiing the previous equation with 119910119895119910119896 implies that

11990111989121198652

11198652

2119868120572= 0 (77)

Therefore we have 119901 = 0 or 119868120572= 0 If 119901 = 0 then

119865 is 1198622-like metric But if 119901 = 0 then 119868120572= 0 that is

1198652is Riemannian metric In this case with similar way

we conclude that 1198651is Riemannian metric But definition

119865 cannot be a Riemannian metric Therefore we have thefollowing

Theorem 13 Every semi-C-reducible twisted product Finslermanifold (119872

1times119891119872

2 119865) is a 1198622-like manifold

4 Riemannian Curvature

The Riemannian curvature of twisted product Finsler man-ifold (119872

1times119891119872

2 119865) with respect to Berwald connection is

given by

R 119886

119887 119888119889=120575119905F119886

119887119888

120575119905x119889minus120575119905F119886

119887119889

120575119905x119888+ F119886

119889119890F119890119887119888minus F119886

119888119890F119890119887119889 (78)

Lemma 14 Let (1198721times119891119872

2 119865) be a twisted product Finsler

manifold Then one has

R119886

119888119889= y119887R 119886

119887 119888119889 (79)

where R119886

119888119889and y119887R 119886

119887 119888119889are given by (50) and (78)

Proof By using (78) we have

y119887R 119894

119887 119896119897= y119887120575119905F119894

119887119896

120575119905x119897minus y119887120575119905F119894

119887119897

120575119905x119896+ y119887F119894

119897119890F119890119887119896 minus y119887F119894

119896119890F119890119887119897 (80)

By using Corollary 8 and Lemma 9 we obtain

y119887120575119905F119894

119887119896

120575119905x119897=120575119905G119894

119896

120575119905119909119897+ F119894

119895119896G119895

119897+ F119894

120573119896G120573

119897

y119887F119894119897119890F119890119887119896= F119894

119897ℎGℎ

119896+ F119894

119897120574G120574

119896

(81)

Interchanging 119894 and 119895 in the previous equation implies that

y119887120575119905F119894

119887119897

120575119905x119896=120575119905G119894

119897

120575119905119909119896+ F119894

119895119897G119895

119896+ F119894

120573119897G120573

119896

y119887F119894119896119890F119890119887119897= F119894

119896ℎGℎ

119897+ F119894

119896120574G120574

119897

(82)

Setting (81) and (82) in (80) gives us y119887R119894

119887 119896119897= R119894

119896119897 In the

similar way we can obtain this relation for another indices

Using (78) we can compute the Riemannian curvature ofa twisted product Finsler manifold

Lemma 15 Let (1198721times119891119872

2 119865) be a twisted product Finsler

manifoldThen the coefficients of Riemannian curvature are asfollows

R119894

119895 119896119897= 119877

119894

119895 119896119897

minus 119872119903

119897

120597119865119894

119895119896

120597119910119903+

120575119905119872

119894

119895119896

120575119905119909119897+119865

119894

119897ℎ119872

ℎ

119895119896+119872

119894

119897ℎ119865ℎ

119895119896minus119872

119894

119897ℎ119872

ℎ

119895119896

+ 119891minus2119892120572120574G119903

120572G119898

120574119862119894

119897119903119862119895119896119898 minusC

119896

119897

(83)

R 119894

120572 119896119897= minus

120575119905

120575119905119909119897(G119903

120572119862119894

119896119903) minus (119865

119894

119903119897minus119872

119894

119903119897)G119898

120572119862119903

119896119898

minus119891minus1G119903

120573119862119894

119897119903119891119896120575120573

120572+ G119903

120573G120583

119896119862119894

119897119903119862120573

120572120583 minus C

119896

119897

(84)

R 119894

119895 120573120582= minus

120575119905

120575119905119906120582(G119903

120573119862119894

119895119903) + G119898

120582G119897

120573119862119894

119903119898119862119903

119895119897

minus (119891119894119892120572120582minus 119891G120583

ℎ119892119894ℎ119862120572120582120583) (119891

119895120575120572

120573minus 119891G]

119895119862120572

120573])

minus C120573

120582

(85)

8 ISRN Geometry

R 119894

120572 120573119897=120575119905

120575119905119906120573(G119903

120572119862119894

119897119903) minus120575119905

120575119905119909119897119891 (119891

119894119892120572120573minus 119891G120582

ℎ119892119894ℎ119862120572120573120582)

minus G119898

120573G119904

120572119862119894

119903119898119862119903

119897119904+ (119891

119894119892120583120573minus 119891G120582

ℎ119892119894ℎ119862120583120573120582)

times (119891119897120575120583

120572minus 119891G]

119897119862120583

120572]) minus 119891119892119903ℎ(119865

119894

119903119897minus119872

119894

119903119897)

times (119891ℎ119892120572120573minus119891G120582

ℎ119862120572120573120582)minusG119903

120583119862119894

119897119903(119865

120583

120572120573+ 119873

120583

120572120573minus119872

120583

120572120573)

(86)

R 119894

119895 120573119897= minus

120575119905

120575119905119909119897(G119903

120573119862119894

119895119903) minus

120575119905

120575119905119906120573(119865

119894

119895119897minus119872

119894

119895119897)

minus (119865119894

119897119903minus119872

119894

119897119903)G119904

120573119862119903

119895119904minus 119891

minus1G119903

120572119862119894

119897119903

times (119891119895120575120572

120573minus 119891G120583

119895119862120572

120573120583) + G119904

120573119862119894

119903119904(119865

119903

119895119897minus119872

119903

119895119897)

+ 119891minus1G119903

120583119862119895119897119903(119891

119894120575120583

120573minus 119891G120582

ℎ119892119894ℎ119862120583

120573120582)

(87)

R 119894

120572 120573120582= minus

120575119905

120575119905119906120582(119891119891

119894119892120572120573minus 119891

2119892119894ℎG120583

ℎ119862120572120573120583) + 119891G119904

120582119862119894

119903119904

times(119891119903119892120572120573minus119891G120583

119897119862120572120573120583119892119903119897)minus119891 (119891

119894119892120582120583minus119891119892

119894ℎG120581

ℎ119862120582120583120581)

times (119865120583

120572120573+ 119873

120583

120572120573minus119872

120583

120572120573) minus C

120573

120582

(88)

R 120574

119895 119896119897=

120575119905

120575119905119909119897(119891

minus2119892120574120582G119903

120582119862119895119896119903) + 119891

minus2119892120574120582G119904

120582119862119897119903119904

times (119865119903

119895119896minus119872

119903

119895119896) + 119891

minus3G119903

120583119862119895119896119903(119891

119897119892120574120583minus 119891G120572

119897119862120574120583

120572)

minus C119896

119897

(89)

R 120574

119895 120573119897=120575119905

120575119905119909119897(119891

minus1119891119895120575120574

120573minus 119891G120572

119895119862120574

120572120573)

minus120575119905

120575119905119906120573(119891

minus2119892120574120582G119903

120582119862119895119897119903) minus 119891

minus2119892120574120582G119904

120582G119898

120573119862ℎ

119897119904119862ℎ119895119898

+ 119891minus2(119891

119897120575120574

120583minus 119891G120572

119897119862120574

120583120572) (119891

119895120575120583

120573minus 119891G]

119895119862120583

120573])

minus 119891minus1(119891

119903120575120574

120573minus 119891G120572

119903119862120574

120573120572) (119865

119903

119895119897minus119872

119903

119895119897)

minus 119891minus2119892120583120582G119903

120582119862119895119897119903(119865

120574

120573120583+ 119873

120574

120573120583minus119872

120574

120573120583)

(90)

R 120574

120572 120573119897=120575119905

120575119905119909119897(119865

120574

120572120573+ 119873

120574

120572120573minus119872

120574

120572120573)

minus120575119905

120575119905119906120573(119891

minus1119891119897120575120574

120572minus G120583

119897119862120574

120572120583)

minus 119891minus1119892120574120582G119904

120582119862ℎ

119897119904(119891

ℎ119892120572120573minus 119891G120583

ℎ119862120572120573120583)

+ 119891minus1(119891

119897120575120574

120583minus 119891G120581

119897119862120574

120583120581) (119865

120583

120572120573+ 119873

120583

120572120573minus119872

120583

120572120573)

+ 119891minus1G119904

120572119862119903

119897119904(119891

119903120575120574

120573minus 119891G120581

119903119862120574

120573120581)

minus 119891minus1(119891

119897120575120583

120572minus 119891G120581

119897119862120583

120572120581) (119865

120574

120573120583+ 119873

120574

120573120583minus119872

120574

120573120583)

(91)

R 120574

119895 120573120582=

120575119905

120575119905119906120582(119891

minus1119891119895120575120574

120573minus G120572

119895119862120574

120572120573)

+ 119891minus1(119865

120574

120572120582+ 119873

120574

120572120582minus119872

120574

120572120582) (119891

119895120575120572

120573minus 119891G]

119895119862120572

120573])

minus119891minus1G119898

120573119862119903

119895119898(119891

119903120575120574

120582minus 119891G120572

119903119862120574

120582120572) minus C

120573

120582

(92)

R 120574

120572 119896119897=

120575119905

120575119905119909119897(119891

minus1119891119896120575120574

120572minus G120583

119896119862120574

120572120583)

+ 119891minus2(119891

119897120575120574

120573minus 119891G120581

119897119862120574

120573120581) (119891

119896120575120573

120572minus 119891G]

119896119862120573

120572])

minus119891minus2G119904

120583G119898

120572119892120574120583119862ℎ

119897119904119862ℎ119896119898 minus C

119896

119897

(93)

R 120574

120572 120573120582= 119877

120574

120572 120573120582minus 119872

120581

120582

120597119865120574

120572120573

120597V120581+

120575119905119872

120574

120572120573

120575119905119906120582+119865

120574

120582120583119872

120583

120572120573+119872

120574

120582120583119865120583

120572120573

minus119872120574

120582120583119872

120583

120572120573+120575119905119873

120574

120572120582

120575119905119906120573+ 119865

120574

120573120583119873

120583

120572120582

+ 119873120574

120573120583119865120583

120572120582+ 119873

120574

120573120583119873

120583

120572120582+ 119873

120574

120582120583119872

120583

120572120573

+ 119873120583

120572120573119872

120574

120582120583+ (119892

119903119904120575120574

120582119891119904minus 119891119892

119903119904G120581

119904119862120574

120582120581)

times (119892120572120573119891119903minus 119891G120583

119903119862120572120573120583) minusC

120573

120582

(94)

where

119872119894

119895119896= 119872

119903

119896119862119894

119895119903+119872

119903

119895119862119894

119896119903minus119872

119903

ℎ119892119894ℎ119862119895119896119903

119872120574

120572120573= 119872

120583

120573119862120574

120572120583+119872

120583

120572119862120574

120573120583minus119872

120583

] 119892120574]119862120572120573120583

119873120574

120572120573= 119891

minus1(119891

120573120575120574

120572+ 119891

120572120575120574

120573minus 119891

120582119892120574120582119892120572120573)

(95)

and C119894

119895denotes the interchange of indices 119894 119895 and subtraction

ByTheorem 18 we have the following

Theorem 16 Let (1198721times119891119872

2 119865) be a flat twisted product

Finsler manifold and let (1198721 119865

1) be Riemannian If 119891 is a

function on1198722 only then (119872

1 119865

1) is locally flat

Similarly we get the following

Theorem 17 Let (1198721times119891119872

2 119865) be a flat twisted product

Finsler manifold and let (1198722 119865

2) be Riemannian If 119891 is a

function on 1198721 only then (119872

2 119865

2) is a space of positive

constant curvature ||119892119903119886119889119891||2

ISRN Geometry 9

Proof Since 1198722is Riemannain and 119891 is a function on 119872

1

then by (94) we obtain

R 120574

120572 120573120582= 119877

120574

120572 120573120582+1003817100381710038171003817119892119903119886119889119891

1003817100381710038171003817

2

(120575120574

120582119892120572120573minus 120575

120574

120573119892120572120582) (96)

Since (1198721times119891119872

2 119865) is flat then R120574

120572 120573120582= 0 Thus the proof is

complete

Theorem 18 Let (1198721times119891119872

2 119865) be a twisted product Rieman-

nian manifold and let 119891 be a function on 1198722 only Then

(1198721times119891119872

2 119865) is flat if and only if (119872

1 119865

1) is flat and the

Riemannian curvature of (1198722 119865

2) satisfies in the following

equation

119877120574

120572 120573120582= 120575119905119873

120574

120572120582

120575119905119906120573+ 119865

120574

120573120583119873

120583

120572120582+ 119873

120574

120573120583119865120583

120572120582+ 119873

120574

120573120583119873

120583

120572120582 minus C

120573

120582

(97)

5 Twisted Product Finsler Manifolds withNon-Riemannian Curvature Properties

There are several important non-Riemannian quantities suchas the Berwald curvature B the mean Berwald curvatureE and the Landsberg curvature L [22] They all vanishfor Riemannian metrics hence they are said to be non-Riemannian In this section we find some necessary and suf-ficient conditions underwhich a twisted product Riemannianmanifold are Berwaldian of isotropic Berwald curvatureof isotropic mean Berwald curvature First we prove thefollowing

Lemma 19 Let (1198721times119891119872

2 119865) be a twisted product Finsler

manifold Then the coefficients of Berwald curvature are asfollows

B120574

120572120573120582= 119861

120574

120572120573120582+ 119891

minus1(119862

120574]120582120572120573119891]119865

2

2+ 2119862

120574]120572120573119891]V120582

+ 2119862120574]120572120582119891]V120573 + 2119862

120574]120572119891]119892120582120573

+ 2119862120574]120582120573119891]V120572 + 2119862

120574]120573119891]119892120582120572

+ 2119862120574]120582119891]119892120572120573 minus 2119862120572120573120582119891

120574)

(98)

B119896

119894119895119897= 119861

119896

119894119895119897+ 119891119862

119896ℎ

119897119895119894119891ℎ1198652

2 (99)

B119896

119894120573119897= 2119891119862

119896ℎ

119894119897119891ℎV120573 (100)

B119896

120572120573119897= 2119891119892

120572120573119862119896ℎ

119897119891ℎ (101)

B119896

120572120573120582= minus 2119891119862

120572120573120582119891119896 (102)

B120574

119894120573120582= B120574

119894119895120582= B120574

119894119895119896= 0 (103)

Let (1198721times119891119872

2 119865) is a Berwald manifold Then we have

B119889

119886119887119888= 0 By using (102) we get

119862120572120573120582119891119896= 0 (104)

Multiplying this equation in 119892119896119903 we obtain

119862120572120573120582119891119903= 0 (105)

Thus if 119891 is not constant on1198721 then we have 119862

120572120573120582= 0 Also

from (101) we result that

119862119896ℎ

119897119891ℎ= 0 (106)

Differentiating this equation with respect to 119910119895 gives us

119862119896ℎ

119897119895119891ℎ= 0 (107)

Similarly we obtain

119862119896ℎ

119897119895119894119891ℎ= 0 (108)

Setting the last equation in (99) implies that 119861119896119894119895119897= 0 that

is (1198721 119865

1) is Berwaldian These explanations give us the

following theorem

Theorem 20 Let (1198721times119891119872

2 119865) be a twisted product Finsler

manifold and let119891 be not constant on1198721Then (119872

1times119891119872

2 119865)

is Berwaldian if and only if (1198721 119865

1) is Berwaldian (119872

2 119865

2) is

Riemannian and the equation 119862119896ℎ119897119891ℎ= 0 is hold

But if 119891 is constant on1198721 that is 119891

119894= 0 then we get the

following

Theorem 21 Let (1198721times119891119872

2 119865) be a twisted product Finsler

manifold and 119891 is constant on 1198721 Then (119872

1times119891119872

2 119865)

is Berwaldian if and only if (1198721 119865

1) is Berwaldian and

the Berwald curvature of (1198722 119865

2) satisfies in the following

equation

119861120574

120572120573120582= minus 119891

minus1(119862

120574]120573120572120582119891]119865

2

2+ 2119862

120574]120573120572119891]V120582 + 2119862

120574]120582120572119891]V120573

+ 2119862120574]120572119891]119892120582120573 + 2119862

120574]120573120582119891]V120572

+ 2119862120574]120573119891]119892120582120572 + 2119862

120574]120582119891]119892120572120573

minus 2119892120574]119862120572120573120582119891])

(109)

Here we consider twisted product Finsler manifold(119872

1times119891119872

2 119865) of isotropic Berwald curvature

Theorem 22 Every isotropic Berwald twisted product Finslermanifold (119872

1times119891119872

2 119865) is a Berwald manifold

Proof Let (1198721times119891119872

2 119865) be an isotropic Berwald manifold

Then we have

B119889

119886119887119888= 119888119865

minus1h119889

119886h119887119888+ h119889

119887h119886119888+ h119889

119888h119886119887+ 2C

119886119887119888y119889 (110)

where 119888 = 119888(x) is a function on119872 Setting 119886 = 119895 119887 = 119896 119888 = 119897and 119889 = 120574 and using (103) imply that

119888119865minus13

1198652119910119895119910119896119910119897V120574 minus V120574 (119910

119895119892119896119897+ 119910

119896119892119895119897+ 119910

119897119892119895119896) = 0 (111)

Multiplying the previous equation in 119910119895119910119896 we derive that119888119891

21198652

11198652

2= 0 Thus we have 119888 = 0 that is (119872

1times119891119872

2) is

Berwaldian

10 ISRN Geometry

Now we are going to study twisted product Finslermanifold of isotropicmean Berwald curvature For this workwe must compute the coefficients of mean Berwald curvatureof a twisted product Finsler manifold

Lemma 23 Let (1198721times119891119872

2 119865) be a twisted product Finsler

manifold Then the coefficients of mean Berwald curvature areas follows

E120572120573= 119864

120572120573+ 119891119892

120572120573119868ℎ119891ℎ+1

2119891119868

]120572120573119891]119865

2

2+ 119891

minus1119891]

times (119862120574]120572120573

V120574+ 119868

]120572V120573+ 119868

]120573V120572+ 119862

]120572120573+ 119868

]119892120572120573)

(112)

E119894119895= 119864

119894119895+1

2119891119868

ℎ

119895119894119891ℎ1198652

2 (113)

E119894120573= 119891119868

ℎ

119894119891ℎV120573 (114)

where 119864119894119895

and 119864120572120573

are the coefficients of mean Berwaldcurvature of (119872

1 119865

1) and (119872

2 119865

2) respectively

Proof By definition and Lemma 19 we get the proof

Theorem24 The twisted product Finslermanifold (1198721times119891119872

2

119865) is weakly Berwald if and only if (1198721 119865

1) is weakly Berwald

119868ℎ119891ℎ= 0 and the following hold

119864120572120573= minus

1

2119891119868

]120572120573119891]119865

2

2minus 119891

minus1119891]

times (119862120574]120572120573

V120574+ 119868

]120572V120573+ 119868

]120573V120572+ 119862

]120572120573+ 119868

]119892120572120573)

(115)

Proof If (1198721times119891119872

2) be a weakly Berwald manifold then we

have

E120572120573= E

119894119895= E

119894120573= 0 (116)

Thus by using (114) we result that 119868ℎ119894119891ℎ= 0 This equation

implies that

119868ℎ

119895119894119891ℎ= 0 119868

ℎ119891ℎ= 0 (117)

By setting these equations in (112) and (113) we conclude that119864119894119895= 0 and 119864

120572120573satisfies in (115)

Now if 119891 is constant on1198722 then (115) implies that 119864

120572120573=

0 Thus we conclude the following

Corollary 25 Let (1198721times119891119872

2 119865) be a twisted product Finsler

manifold and let 119891 be a function on 1198721 only Then

(1198721times119891119872

2 119865) is weakly Berwald if and only if (119872

1 119865

1) and

(1198722 119865

2) are weakly Berwald manifolds and 119868ℎ119891

ℎ= 0

Now we consider twisted product Finsler manifolds withisotropic mean Berwald curvature It is remarkable that as aconsequence of Lemma 23 we have the following

Lemma26 Twisted product Finslermanifold (1198721times119891119872

2 119865) is

isotropic mean Berwald manifold if and only if

119864120572120573+ 119891119892

120572120573119868ℎ119891ℎ+1

2119891119868

]120572120573119891]119865

2

2+ 119891

minus1119891]

times (119862120574]120572120573

V120574+ 119868

]120572V120573+ 119868

]120573V120572+ 119862

]120572120573+ 119868

]119892120572120573)

minus119899 + 1

2119888119891

2119865minus1(119892

120572120573minus1198912

1198652V120572V120573) = 0

(118)

119864119894119895+1

2119891119868

ℎ

119895119894119891ℎ1198652

2minus119899 + 1

2119888119865

minus1(119892

119894119895minus1

1198652119910119894119910119895) = 0 (119)

119888 (119899 + 1) 119865minus3119910119894+ 119891119868

ℎ

119894119891ℎ= 0 (120)

where 119888 = 119888(x) is a scalar function on119872

Theorem 27 Every twisted product Finsler manifold(119872

1times119891119872

2 119865) with isotropic mean Berwald curvature is a

weakly Berwald manifold

Proof Suppose that 119865 is isotropic mean Berwald twistedproduct Finsler metric Then differentiating (120) withrespect to V120574 gives us

119888 (119899 + 1) 1198912119865minus5V

120574119910119894= 0 (121)

Thus we conclude that 119888 = 0 This implies that 119865 reduces to aweakly Berwald metric

6 Locally Dually Flat Twisted ProductFinsler Manifolds

In [23] Amari and Nagaoka introduced the notion of duallyflat Riemannian metrics when they study the informationgeometry on Riemannian manifolds Information geometryhas emerged from investigating the geometrical structure ofa family of probability distributions and has been appliedsuccessfully to various areas including statistical inferencecontrol system theory andmultiterminal information theoryIn Finsler geometry Shen extends the notion of locally duallyflatness for Finsler metrics [17] Dually flat Finsler metricsform a special and valuable class of Finsler metrics in Finslerinformation geometry which play a very important role instudying flat Finsler information structure [24 25]

In this section we study locally dually flat twisted productFinsler metrics It is remarkable that a Finsler metric 119865 =119865(x y) on a manifold119872 is said to be locally dually flat if atany point there is a standard coordinate system (x119886 y119886) in119879119872such that it satisfies

12059721198652

120597x119887120597y119886y119887 = 2120597119865

2

120597x119886 (122)

In this case the coordinate (x119886) is called an adapted localcoordinate system Byusing (122) we can obtain the followinglemma

ISRN Geometry 11

Lemma 28 Let (1198721times119891119872

2 119865) be a twisted product Finsler

manifold Then 119865 is locally dually flat if and only if 1198651and 119865

2

satisfy in the following equations

12059721198652

1

120597119909119896120597119910119897119910119896= 2120597119865

2

1

120597119909119897+ 4119891119891

1198971198652

2 (123)

4119891119896V120573119910119896+ 119891

12059721198652

2

120597119906120572120597V120573V120572 + 4119891

120572V120573V120572 = 2119891

1205971198652

2

120597119906120573+ 4119891

1205731198652

2

(124)

Now let 119865 be a locally dually flat Finsler metric Takingderivative with respect to V120574 from (123) yields 119891

119897= 0 which

means that 119891 is a constant function on1198721 In this case the

relations (123) and (124) reduce to the following

12059721198652

1

120597119909119896120597119910119897119910119896= 2120597119865

2

1

120597119909119897 (125)

11989112059721198652

2

120597119906120572120597V120573V120572 + 4119891

120572V120573V120572 = 2119891

1205971198652

2

120597119906120573+ 4119891

1205731198652

2 (126)

By (125) we deduce that 1198651is locally dually flat

Now we assume that 1198651and 119865

2are locally dually flat

Finsler metrics Then we have

12059721198652

1

120597119909119896120597119910119897119910119896= 2120597119865

2

1

120597119909119897

12059721198652

2

120597119906120572120597V120573V120572 = 2

1205971198652

2

120597119906120573

(127)

By (127) we derive that (123) and (124) are hold if and only ifthe following hold

119891119897= 0 119891

120572V120573V120572 = 119891

1205731198652

2 (128)

Therefore we can conclude the following

Theorem 29 Let (1198721times119891119872

2 119865) be a twisted product Finsler

manifold

(i) If 119865 is locally dually flat then 1198651is locally dually flat 119891

is a function with respect to (119906120572) only and 1198652satisfies

in (126)(ii) If 119865

1and 119865

2are locally dually flat then 119865 is locally

dually flat if and only if 119891 is a function with respect(119906

120572) only and 119865

2satisfies in (128)

ByTheorem 29 we conclude the following

Corollary 30 There is not exist any locally dually flat propertwisted product Finsler manifold

References

[1] J K Beem P E Ehrlich and K L Easley Global LorentzianGeometry vol 202 of Monographs and Textbooks in Pure andApplied Mathematics Marcel Dekker New York NY USA 2ndedition 1996

[2] G Ganchev and V Mihova ldquoRiemannian manifolds of quasi-constant sectional curvaturesrdquo Journal fur die Reine und Ange-wandte Mathematik vol 522 pp 119ndash141 2000

[3] B OrsquoNeill Semi-Riemannian Geometry With Applications toRelativity vol 103 of Pure and Applied Mathematics AcademicPress New York NY USA 1983

[4] M Lohnherr and H Reckziegel ldquoOn ruled real hypersurfacesin complex space formsrdquo Geometriae Dedicata vol 74 no 3pp 267ndash286 1999

[5] B-Y Chen F Dillen L Verstraelen and L VranckenldquoLagrangian isometric immersions of a real-space-form119872119899

(119888)

into a complex-space-form119872119899(4119888)rdquoMathematical Proceedings

of the Cambridge Philosophical Society vol 124 no 1 pp 107ndash125 1998

[6] N Koike ldquoThe decomposition of curvature netted hypersur-facesrdquo Geometriae Dedicata vol 54 no 1 pp 1ndash11 1995

[7] B-y Chen Geometry of Submanifolds and its ApplicationsScience University of Tokyo Tokyo Japan 1981

[8] R Ponge and H Reckziegel ldquoTwisted products in pseudo-Riemannian geometryrdquo Geometriae Dedicata vol 48 no 1 pp15ndash25 1993

[9] B-Y Chen ldquoTwisted product CR-submanifolds in Kaehlermanifoldsrdquo Tamsui Oxford Journal of Mathematical Sciencesvol 16 no 2 pp 105ndash121 2000

[10] L Kozma I R Peter and H Shimada ldquoOn the twisted productof Finsler manifoldsrdquo Reports on Mathematical Physics vol 57no 3 pp 375ndash383 2006

[11] M Matsumoto and C Shibata ldquoOn semi-C-reducibility T-tensor= 0 and S4-likeness of Finsler spacesrdquo Journal of Math-ematics of Kyoto University vol 19 no 2 pp 301ndash314 1979

[12] B Najafi Z Shen and A Tayebi ldquoFinsler metrics of scalar flagcurvature with special non-Riemannian curvature propertiesrdquoGeometriae Dedicata vol 131 pp 87ndash97 2008

[13] E Peyghan andAHeydari ldquoConformal vector fields on tangentbundle of a Riemannian manifoldrdquo Journal of MathematicalAnalysis and Applications vol 347 no 1 pp 136ndash142 2008

[14] Z Shen Differential Geometry of Spray and Finsler SpacesKluwer Academic Publishers Dordrecht The Netherlands2001

[15] M Matsumoto and S-I Hojo ldquoA conclusive theorem on C-reducible Finsler spacesrdquo The Tensor Society vol 32 no 2 pp225ndash230 1978

[16] D Bao S-S Chern and Z Shen An Introduction to Riemann-Finsler Geometry vol 200 of Graduate Texts in MathematicsSpringer New York NY USA 2000

[17] Z Shen ldquoRiemann-Finsler geometry with applications to infor-mation geometryrdquo Chinese Annals of Mathematics B vol 27 no1 pp 73ndash94 2006

[18] Y Ichijyo ldquoFinsler manifolds modeled on a Minkowski spacerdquoJournal of Mathematics of Kyoto University vol 16 no 3 pp639ndash652 1976

[19] X Chen and Z Shen ldquoOn Douglas metricsrdquo PublicationesMathematicae Debrecen vol 66 no 3-4 pp 503ndash512 2005

[20] L Berwald ldquoUber Parallelubertragung in Raumen mit allge-meiner Massbestimmungrdquo Jahresbericht der DMVmdashDeutscheMathematiker-Vereinigung vol 34 pp 213ndash220 1926

[21] M Matsumoto ldquoOn Finsler spaces with Randersrsquo metric andspecial forms of important tensorsrdquo Journal of Mathematics ofKyoto University vol 14 pp 477ndash498 1974

12 ISRN Geometry

[22] A Tayebi and E Peyghan ldquoOn special Berwald metricsrdquoSymmetry Integrability and Geometry vol 6 article 008 9pages 2010

[23] S-I Amari and H NagaokaMethods of Information Geometryvol 191 of Translations of Mathematical Monographs AmericanMathematical Society Providence RI USA 2000

[24] X Cheng Z Shen and Y Zhou ldquoOn locally dually flat Finslermetricsrdquo International Journal ofMathematics vol 21 no 11 pp1531ndash1543 2010

[25] Q Xia ldquoOn a class of locally dually flat Finsler metrics ofisotropic flag curvaturerdquo Publicationes Mathematicae Debrecenvol 78 no 1 pp 169ndash190 2011

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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

4 ISRN Geometry

Lemma 1 The coefficients G119886

119887defined by (23) satisfy in the

following

(G119886

119887(119909 119906 119910 V)) = [

G119894

119895(119909 119906 119910 V) G120572

119895(119909 119906 119910 V)

G119894

120573(119909 119906 119910 V) G120572

120573(119909 119906 119910 V)] (24)

where

G119894

119895(119909 119906 119910 V) =

120597G119894

120597119910119895= 119866

119894

119895+ 119862

119894ℎ

119895119891119891

ℎ1198652

2 (25)

G119894

120573(119909 119906 119910 V) =

120597G119894

120597V120573= minus119891119891

119894V120573 (26)

G120572

119895(119909 119906 119910 V) =

120597G120572

120597119910119895= 119891

minus1119891119895V120572 (27)

G120572

120573(119909 119906 119910 V) =

120597G120572

120597V120573

= 119866120572

120573+ 119891

minus1(119862

120572120574

1205731198911205741198652

2+ 119891

119895119910119895120575120572

120573

minus 119891120572V

120573+ 119891

120573V120572 + 119891

120574V120574120575120572

120573)

(28)

Next 119881119879119872∘ kernel of the differential of the projectionmap

120587 = (1205871 120587

2) 119879119872

∘

1oplus 119879119872

∘

2997888rarr 119872

1times119872

2 (29)

which is a well-defined subbundle of 119879119879119872∘ is consid-ered Locally Γ(119881119879119872∘

) is spanned by the natural vectorfields 1205971205971199101 1205971205971199101198991 120597120597V1 120597120597V1198992 and it is calledthe twisted vertical distribution on 119879119872∘ Then using thefunctions given by (25)ndash(28) the nonholonomic vector fieldsare defined as follows

120575119905

120575119905119909119894=120597

120597119909119894minus G119895

119894

120597

120597119910119895minus G120573

119894

120597

120597V120573 (30)

120575119905

120575119905119906120572=120597

120597119906120572minus G119895

120572

120597

120597119910119895minus G120573

120572

120597

120597V120573 (31)

which make it possible to construct a complementary vectorsubbundle119867119879119872∘ to 119881119879119872∘ in 119879119879119872∘ as follows

119867119879119872∘= span 120575

119905

1205751199051199091

120575119905

1205751199051199091198991120575119905

1205751199051199061

120575119905

1205751199051199061198992 (32)

119867119879119872∘ is called the twisted horizontal distribution on 119879119872∘

Thus the tangent bundle of 119879119872∘ admits the decomposition

119879119879119872∘= 119867119879119872

∘oplus 119881119879119872

∘ (33)

It is shown thatG = (G119886

119887) is a nonlinear connection on119879119872 =

1198791198721oplus 119879119872

2 In the following we compute the nonlinear

connection of a twisted product Finsler manifold

Proposition 2 If (1198721times119891119872

2 119865) is a twisted product Finsler

manifold then G = (G119886

119887) is the nonlinear connection on 119879119872

Further one has

120597G119894

119895

120597119910119896119910119896+

120597G119894

119895

120597V120574V120574 = G119894

119895

120597G119894

120573

120597119910119896119910119896+

120597G119894

120573

120597V120574V120574 = G119894

120573

120597G120572

119895

120597119910119896119910119896+

120597G120572

119895

120597V120574V120574 = G120572

119895

120597G120572

120573

120597119910119896119910119896+

120597G120572

120573

120597V120574V120574 = G120572

120573

(34)

Definition 3 Using decomposition (33) the twisted verticalmorphism V119905 119879119879119872∘

rarr 119881119879119872∘ is defined by

V119905 =120597

120597119910119894otimes 120575

119905119910119894+120597

120597V120572otimes 120575

119905V120572 (35)

where

120575119905119910119894= 119889119910

119894+ G119894

119895119889119909

119895+ G119894

120573119889119906

120573

120575119905V120572 = 119889V120572 + G120572

119895119889119909

119895+ G120572

120573119889119906

120573

(36)

For this projective morphism the following hold

V119905 (120597

120597119910119894) =

120597

120597119910119894 V119905 (

120597

120597V120572) =

120597

120597V120572

V119905 (120575119905

120575119905119909119894) = 0 V119905 (

120575119905

120575119905119906119894) = 0

(37)

From the previous equations we conclude that

(V119905)2

= V119905 ker (V119905) = 119867119879119872∘ (38)

This mapping is called the twisted vertical projective

Definition 4 Using decomposition (33) the doubly warpedhorizontal projective ℎ119905 119879119879119872∘

rarr 119867119879119872∘ is defined by

ℎ119905= 119894119889 minus V119905 (39)

or

ℎ119905=120575119905

120575119905119909119894otimes 119889119909

119894+120575119905

120575119905119906120572otimes 119889119906

120572 (40)

For this projective morphism the following hold

ℎ119905(120575119905

120575119905119909119894) =

120575119905

120575119905119909119894 ℎ

119905(120575119905

120575119905119906120572) =

120575119905

120575119905119906120572

ℎ119905(120597

120597119910119894) = 0 ℎ

119905(120597

120597V120572) = 0

(41)

Thus we result that

(ℎ119905)2

= ℎ119905 ker (ℎ119905) = 119881119879119872∘

(42)

ISRN Geometry 5

Definition 5 Using decomposition (33) the twisted almosttangent structure 119869119905 119867119879119872∘

rarr 119881119879119872∘ is defined by

119869119905120597

120597119910119894otimes 119889119909

119894+120597

120597V120572otimes 119889119906

120572 (43)

or

119869119905(120575119905

120575119905119909119894) =

120597

120597119910119894 119869

119905(120575119905

120575119905119906120572) =

120597

120597V120572

119869119905(120597

120597119910119894) = 119869

119905(120597

120597V120572) = 0

(44)

Thus we result that

(119869119905)2

= 0 ker 119869119905 = 119868119898119869119905 = 119881119879119872∘ (45)

Here we introduce some geometrical objects of twistedproduct Finsler manifold In order to simplify the equationswe rewritten the basis of119867119879119872∘ and 119881119879119872∘ as follows

120575119905

120575119905x119886=120575119905

120575119905119909119894120575119894

119886+120575119905

120575119905119906120572120575120572

119886

120597

120597y119886=120597

120597119910119894120575119894

119886+120597

120597V120572120575120572

119886

(46)

Thus

119879119879119872∘= span 120575

119905

120575119905x119886120597

120597y119886 (47)

The Lie brackets of this basis is given by

[120575119905

120575119905x119886120575119905

120575119905x119887] = R119888

119886119887

120597

120597y119888

[120575119905

120575119905x119886120597

120597y119887] = G119888

119886119887

120597

120597y119888

[120597

120597y119886120597

120597y119887] = 0

(48)

where

(i) R119888

119886119887=120575119905G119888

119886

120575119905x119887minus120575119905G119888

119887

120575119905x119886 (49)

(ii) G119888

119886119887=120597G119888

119886

120597y119887 (50)

Therefore we have the following

Corollary 6 Let (1198721times119891119872

2 119865) be a twisted product Finsler

manifold Then

R119888

119886119887= (R119896

119894119895R119896

119894120573R119896

120572119895R119896

120572120573R120574

119894119895R120574

119894120573R120574

120572119895R120574

120572120573)

(51)

where

R119896

119894119895=120575119905G119896

119894

120575119905119909119895minus

120575119905G119896

119895

120575119905119909119894 R119896

119894120573=120575119905G119896

119894

120575119905119906120573minus

120575119905G119896

120573

120575119905119909119894

R119896

120572119895=120575119905G119896

120572

120575119905119909119895minus

120575119905G119896

119895

120575119905119906120572 R119896

120572120573=120575119905G119896

120572

120575119905119906120573minus

120575119905G119896

120573

120575119905119906120572

R120574

119894119895=120575119905G120574

119894

120575119905119909119895minus

120575119905G120574

119895

120575119905119909119894 R120574

119894120573=120575119905G120574

119894

120575119905119906120573minus

120575119905G120574

120573

120575119905119909119894

R120574

120572119895=120575119905G120574

120572

120575119905119909119895minus

120575119905G120574

119895

120575119905119906120572 R120574

120572120573=120575119905G120574

120572

120575119905119906120573minus

120575119905G120574

120573

120575119905119906120572

(52)

With a simple calculation we have the following

Corollary 7 Let (1198721times119891119872

2 119865) be a twisted product Finsler

manifold Then

G119888

119886119887= (G119896

119894119895G119896

119894120573G119896

120572119895G119896

120572120573G120574

119894119895G120574

119894120573G120574

120572119895G120574

120572120573) (53)

where

G120574

120572120573=120597G120574

120572

120597V120573

= 119866120574

120572120573+ 119891

minus1(119862

120574120582

1205721205731198911205821198652

2+ 2119862

120574120582

120572119891120582V120573+ 2119862

120574120582

120573119891120582V120572

minus 119891120574119892120572120573+ 119891

120573120575120574

120572+ 119891

120572120575120574

120573) = G120574

120573120572

G119896

119894119895=120597G119896

119894

120597119910119895= 119866

119896

119894119895+ 119862

119896ℎ

119894119895119891119891

ℎ1198652

2= G119896

119895119894

G119896

119894120573=120597G119896

119894

120597V120573= 2119862

119896ℎ

119894119891119891

ℎV120573= G119896

120573119894

G119896

120572120573=120597G119896

120572

120597V120573= minus119891119891

119896119892120572120573= G119896

120573120572

G120574

119894120573=120597G120574

119894

120597V120573= 119891

minus1119891119894120575120574

120573= G120574

120573119894

G120574

119894119895=120597G120574

119894

120597119910119895= G120574

119895119894= 0

(54)

where 119862119896ℎ119894119895= 120597119862

119896ℎ

119894120597119910

119895 Apart from G119888

119886119887 the functions F119888

119886119887are

given by

F119888119886119887=1

2g119888119890 (120575

119905g119890119886

120575119905x119887+120575119905g

119890119887

120575119905x119886minus120575119905g

119886119887

120575119905x119890) (55)

Corollary 8 Let (1198721times119891119872

2 119865) be a twisted product Finsler

manifold Then

F119888119886119887= (F119896

119894119895 F119896

119894120573 F119896

120572119895 F119896

120572120573 F120574

119894119895 F120574

119894120573 F120574

120572119895 F120574

120572120573) (56)

6 ISRN Geometry

where

F119896119894119895= 119865

119896

119894119895minus (119872

119903

119895119862119896

119894119903+119872

119903

119894119862119896

119895119903minus119872

119903

ℎ119862119894119895119903119892119896ℎ) (57)

F119896119894120573= minusG119903

120573119862119896

119894119903= F119896

120573119894 (58)

F119896120572120573= minus119891119891

119896119892120572120573+ 119891

2119892119896ℎG120582

ℎ119862120572120573120582 (59)

F120574119894119895= 119891

minus2119892120574120582G119903

120582119862119894119895119903 (60)

F120574119894120573= 119891

minus1119891119894120575120574

120573minus G120572

119894119862120574

120572120573= F120574

120573119894 (61)

F120574120572120573= 119865

120574

120572120573+ 119873

120574

120572120573minus (119872

120583

120573119862120574

120572120583+119872

120583

120572119862120574

120573120583minus119872

120583

120582119862120572120573120583119892120574120582)

(62)

119865119896

119894119895=1

2119892119896ℎ(120575119892

ℎ119894

120575119909119895+

120575119892ℎ119895

120575119909119894minus

120575119892119894119895

120575119909ℎ)

119865120574

120572120573=1

2119892120574120582(120575119892

120582120572

120575119906120573+

120575119892120582120573

120575119906120572minus

120575119892120572120573

120575119906120582)

119872119903

119894= 119862

119903ℎ

119894119891119891

ℎ1198652

2

119872120583

120572= 119891

minus1(119862

120583120574

1205721198911205741198652

2+ 119891

119903119910119903120575120583

120572+ 119891

120574V120574120575120583

120572minus 119892

120583120574119891120574V120572+ 119891

120572V120583)

119873120574

120572120573= 119891

minus1(119891

120573120575120574

120572+ 119891

120572120575120574

120573minus 119891

120582119892120574120582119892120572120573)

(63)

Proof By using (55) we have

F119896119894119895=1

2119892119896ℎ(120575119905119892ℎ119894

120575119905119909119895+

120575119905119892ℎ119895

120575119905119909119894minus

120575119905119892119894119895

120575119905119909ℎ) (64)

Since 119892119894119895is a function with respect to (119909 119910) then by (25) and

(30) we obtain

120575119905119892ℎ119894

120575119905119909119895=120575119892

ℎ119894

120575119909119895minus 2119872

119903

119895119862ℎ119894119903 (65)

Interchanging 119894 119895 and ℎ in the previous equation gives us

120575119905119892ℎ119895

120575119905119909119894=

120575119892ℎ119895

120575119909119894minus 2119872

119903

119894119862ℎ119895119903

120575119905119892119894119895

120575119905119909ℎ=

120575119892119894119895

120575119909ℎminus 2119872

119903

ℎ119862119894119895119903

(66)

Putting these equation in (64) give us (57) In the similar waywe can prove the another relation

By using (i) of (23) and (57)ndash(62) we can conclude thefollowing

Lemma 9 Let (1198721times119891119872

2 119865) be a twisted product Finsler

manifold Then y119888F119886119887119888= G119886

119887 where F119886

119887119888and G119886

119887are defined by

(55) and (i) of (23) respectively

The Cartan torsion is one of the most important non-Riemannian quantity in Finsler geometry and it is first

introduced by Finsler and emphasized by Cartan whichmeasures a departure from a Riemannian manifold Moreprecisely a Finsler metric reduces to a Riemannian metricif and only if it has vanishing Cartan torsion The localcomponents of Cartan tensor field of the twisted Finslermanifold (119872

1times119891119872

2 119865) is defined by

C119886

119887119888=1

2g119886119890 120597g119887119890120597y119888 (67)

From this definition we conclude the following

Lemma 10 Let119862119896119894119895and119862120574

120572120573be the local components of Cartan

tensor field on1198721and119872

2 respectively Then one has

C119888

119886119887= (C119896

119894119895C119896

119894120573C119896

120572119895C119896

120572120573C120574

119894119895C120574

119894120573C120574

120572119895C120574

120572120573) (68)

where

C119896

119894119895=1

2119892119896ℎ120597119892

119894119895

120597119910ℎ= 119862

119896

119894119895

C120574

120572120573=1

2119892120574120582120597119892

120572120573

120597V120582= 119862

120574

120572120573

(69)

and C119896

119894120573= C119896

120572119895= C119896

120572120573= C120574

119894119895= C120574

119894120573= C120574

120572119895= 0

By using the Lemma 10 we can get the following

Corollary 11 Let (1198721times119891119872

2 119865) be a twisted product Finsler

manifoldThen (1198721times119891119872

2 119865) is a Riemannianmanifold if and

only if (1198721 119865

1) and (119872

2 119865

2) are Riemannian manifold

Various interesting special forms of Cartan tensors havebeen obtained by some Finslerians [11] The Finsler spaceshaving such special forms have been called C-reducible C2-like semi-C-reducible and so forth In [21] Matsumotointroduced the notion of C-reducible Finsler metrics andproved that any Randers metric is C-reducible Later onMatsumoto-Hojo proves that the converse is true too [15]

Here we define the Matsumoto twisted tensorM119886119887119888

for atwisted product Finsler manifold (119872

1times119891119872

2 119865) as follows

M119886119887119888= C

119886119887119888minus1

119899 + 1I

119886h119887119888+ I

119887h119886119888+ I

119888h119886119887 (70)

where I119886= g119887119888C

119886119887119888C

119886119887119888= g

119888119889C119889

119886119887 andh

119886119887= g

119886119887minus(1119865

2)y

119886y119887

By attention to the previous equation and relations

C119894119895119896= 119862

119894119895119896 C

120572120573120574= 119891

2119862120572120573120574 (71)

we obtain

M120572119895119896= minus

1

119899 + 1119868

120572(119892

119895119896minus1

1198652119910119895119910119896) minus1198912

1198652V120572(119868

119895119910119896+ 119868

119896119910119895)

(72)

Contracting the previous equation in 119910119895119910119896 gives us

119910119895119910119896M

120572119895119896= minus11989121198652

11198652

2

(119899 + 1) 1198652119868120572 (73)

ISRN Geometry 7

Similarly we obtain

V120582V120573M119894120573120582= minus11989121198652

11198652

2

(119899 + 1) 1198652119868119894 (74)

Therefore if M119894120573120582= M

120572119895119896= 0 then we get 119868

119894= 119868

120572= 0 that

is (1198721 119865

1) and (119872

2 119865

2) are Riemannian manifolds Thus we

have the following

Theorem 12 There is not exist any C-reducible twisted prod-uct Finsler manifold

Now we are going to consider semi-C-reducible twistedproduct Finsler manifold (119872

1times119891119872

2 119865) Let (119872

1times119891119872

2 119865) be

a semi-C-reducible twisted product Finsler manifold Thenwe have

C119886119887119888=119901

119899 + 1I

119886h119887119888+ I

119887h119886119888+ I

119888h119886119887 +119902

C2I119886I119887I119888 (75)

where C2= I119886I

119886and 119901 and 119902 are scalar function on119872

1times119891119872

2

with 119901 + 119902 = 1 This equation gives us

0 = C120572119895119896

=119901

119899 + 1119868

120572(119892

119895119896minus1

1198652119910119895119910119896) minus1198912

1198652V120572(119868

119895119910119896+ 119868

119896119910119895)

+119902

C2119868120572119868119895119868119896

(76)

Contractiing the previous equation with 119910119895119910119896 implies that

11990111989121198652

11198652

2119868120572= 0 (77)

Therefore we have 119901 = 0 or 119868120572= 0 If 119901 = 0 then

119865 is 1198622-like metric But if 119901 = 0 then 119868120572= 0 that is

1198652is Riemannian metric In this case with similar way

we conclude that 1198651is Riemannian metric But definition

119865 cannot be a Riemannian metric Therefore we have thefollowing

Theorem 13 Every semi-C-reducible twisted product Finslermanifold (119872

1times119891119872

2 119865) is a 1198622-like manifold

4 Riemannian Curvature

The Riemannian curvature of twisted product Finsler man-ifold (119872

1times119891119872

2 119865) with respect to Berwald connection is

given by

R 119886

119887 119888119889=120575119905F119886

119887119888

120575119905x119889minus120575119905F119886

119887119889

120575119905x119888+ F119886

119889119890F119890119887119888minus F119886

119888119890F119890119887119889 (78)

Lemma 14 Let (1198721times119891119872

2 119865) be a twisted product Finsler

manifold Then one has

R119886

119888119889= y119887R 119886

119887 119888119889 (79)

where R119886

119888119889and y119887R 119886

119887 119888119889are given by (50) and (78)

Proof By using (78) we have

y119887R 119894

119887 119896119897= y119887120575119905F119894

119887119896

120575119905x119897minus y119887120575119905F119894

119887119897

120575119905x119896+ y119887F119894

119897119890F119890119887119896 minus y119887F119894

119896119890F119890119887119897 (80)

By using Corollary 8 and Lemma 9 we obtain

y119887120575119905F119894

119887119896

120575119905x119897=120575119905G119894

119896

120575119905119909119897+ F119894

119895119896G119895

119897+ F119894

120573119896G120573

119897

y119887F119894119897119890F119890119887119896= F119894

119897ℎGℎ

119896+ F119894

119897120574G120574

119896

(81)

Interchanging 119894 and 119895 in the previous equation implies that

y119887120575119905F119894

119887119897

120575119905x119896=120575119905G119894

119897

120575119905119909119896+ F119894

119895119897G119895

119896+ F119894

120573119897G120573

119896

y119887F119894119896119890F119890119887119897= F119894

119896ℎGℎ

119897+ F119894

119896120574G120574

119897

(82)

Setting (81) and (82) in (80) gives us y119887R119894

119887 119896119897= R119894

119896119897 In the

similar way we can obtain this relation for another indices

Using (78) we can compute the Riemannian curvature ofa twisted product Finsler manifold

Lemma 15 Let (1198721times119891119872

2 119865) be a twisted product Finsler

manifoldThen the coefficients of Riemannian curvature are asfollows

R119894

119895 119896119897= 119877

119894

119895 119896119897

minus 119872119903

119897

120597119865119894

119895119896

120597119910119903+

120575119905119872

119894

119895119896

120575119905119909119897+119865

119894

119897ℎ119872

ℎ

119895119896+119872

119894

119897ℎ119865ℎ

119895119896minus119872

119894

119897ℎ119872

ℎ

119895119896

+ 119891minus2119892120572120574G119903

120572G119898

120574119862119894

119897119903119862119895119896119898 minusC

119896

119897

(83)

R 119894

120572 119896119897= minus

120575119905

120575119905119909119897(G119903

120572119862119894

119896119903) minus (119865

119894

119903119897minus119872

119894

119903119897)G119898

120572119862119903

119896119898

minus119891minus1G119903

120573119862119894

119897119903119891119896120575120573

120572+ G119903

120573G120583

119896119862119894

119897119903119862120573

120572120583 minus C

119896

119897

(84)

R 119894

119895 120573120582= minus

120575119905

120575119905119906120582(G119903

120573119862119894

119895119903) + G119898

120582G119897

120573119862119894

119903119898119862119903

119895119897

minus (119891119894119892120572120582minus 119891G120583

ℎ119892119894ℎ119862120572120582120583) (119891

119895120575120572

120573minus 119891G]

119895119862120572

120573])

minus C120573

120582

(85)

8 ISRN Geometry

R 119894

120572 120573119897=120575119905

120575119905119906120573(G119903

120572119862119894

119897119903) minus120575119905

120575119905119909119897119891 (119891

119894119892120572120573minus 119891G120582

ℎ119892119894ℎ119862120572120573120582)

minus G119898

120573G119904

120572119862119894

119903119898119862119903

119897119904+ (119891

119894119892120583120573minus 119891G120582

ℎ119892119894ℎ119862120583120573120582)

times (119891119897120575120583

120572minus 119891G]

119897119862120583

120572]) minus 119891119892119903ℎ(119865

119894

119903119897minus119872

119894

119903119897)

times (119891ℎ119892120572120573minus119891G120582

ℎ119862120572120573120582)minusG119903

120583119862119894

119897119903(119865

120583

120572120573+ 119873

120583

120572120573minus119872

120583

120572120573)

(86)

R 119894

119895 120573119897= minus

120575119905

120575119905119909119897(G119903

120573119862119894

119895119903) minus

120575119905

120575119905119906120573(119865

119894

119895119897minus119872

119894

119895119897)

minus (119865119894

119897119903minus119872

119894

119897119903)G119904

120573119862119903

119895119904minus 119891

minus1G119903

120572119862119894

119897119903

times (119891119895120575120572

120573minus 119891G120583

119895119862120572

120573120583) + G119904

120573119862119894

119903119904(119865

119903

119895119897minus119872

119903

119895119897)

+ 119891minus1G119903

120583119862119895119897119903(119891

119894120575120583

120573minus 119891G120582

ℎ119892119894ℎ119862120583

120573120582)

(87)

R 119894

120572 120573120582= minus

120575119905

120575119905119906120582(119891119891

119894119892120572120573minus 119891

2119892119894ℎG120583

ℎ119862120572120573120583) + 119891G119904

120582119862119894

119903119904

times(119891119903119892120572120573minus119891G120583

119897119862120572120573120583119892119903119897)minus119891 (119891

119894119892120582120583minus119891119892

119894ℎG120581

ℎ119862120582120583120581)

times (119865120583

120572120573+ 119873

120583

120572120573minus119872

120583

120572120573) minus C

120573

120582

(88)

R 120574

119895 119896119897=

120575119905

120575119905119909119897(119891

minus2119892120574120582G119903

120582119862119895119896119903) + 119891

minus2119892120574120582G119904

120582119862119897119903119904

times (119865119903

119895119896minus119872

119903

119895119896) + 119891

minus3G119903

120583119862119895119896119903(119891

119897119892120574120583minus 119891G120572

119897119862120574120583

120572)

minus C119896

119897

(89)

R 120574

119895 120573119897=120575119905

120575119905119909119897(119891

minus1119891119895120575120574

120573minus 119891G120572

119895119862120574

120572120573)

minus120575119905

120575119905119906120573(119891

minus2119892120574120582G119903

120582119862119895119897119903) minus 119891

minus2119892120574120582G119904

120582G119898

120573119862ℎ

119897119904119862ℎ119895119898

+ 119891minus2(119891

119897120575120574

120583minus 119891G120572

119897119862120574

120583120572) (119891

119895120575120583

120573minus 119891G]

119895119862120583

120573])

minus 119891minus1(119891

119903120575120574

120573minus 119891G120572

119903119862120574

120573120572) (119865

119903

119895119897minus119872

119903

119895119897)

minus 119891minus2119892120583120582G119903

120582119862119895119897119903(119865

120574

120573120583+ 119873

120574

120573120583minus119872

120574

120573120583)

(90)

R 120574

120572 120573119897=120575119905

120575119905119909119897(119865

120574

120572120573+ 119873

120574

120572120573minus119872

120574

120572120573)

minus120575119905

120575119905119906120573(119891

minus1119891119897120575120574

120572minus G120583

119897119862120574

120572120583)

minus 119891minus1119892120574120582G119904

120582119862ℎ

119897119904(119891

ℎ119892120572120573minus 119891G120583

ℎ119862120572120573120583)

+ 119891minus1(119891

119897120575120574

120583minus 119891G120581

119897119862120574

120583120581) (119865

120583

120572120573+ 119873

120583

120572120573minus119872

120583

120572120573)

+ 119891minus1G119904

120572119862119903

119897119904(119891

119903120575120574

120573minus 119891G120581

119903119862120574

120573120581)

minus 119891minus1(119891

119897120575120583

120572minus 119891G120581

119897119862120583

120572120581) (119865

120574

120573120583+ 119873

120574

120573120583minus119872

120574

120573120583)

(91)

R 120574

119895 120573120582=

120575119905

120575119905119906120582(119891

minus1119891119895120575120574

120573minus G120572

119895119862120574

120572120573)

+ 119891minus1(119865

120574

120572120582+ 119873

120574

120572120582minus119872

120574

120572120582) (119891

119895120575120572

120573minus 119891G]

119895119862120572

120573])

minus119891minus1G119898

120573119862119903

119895119898(119891

119903120575120574

120582minus 119891G120572

119903119862120574

120582120572) minus C

120573

120582

(92)

R 120574

120572 119896119897=

120575119905

120575119905119909119897(119891

minus1119891119896120575120574

120572minus G120583

119896119862120574

120572120583)

+ 119891minus2(119891

119897120575120574

120573minus 119891G120581

119897119862120574

120573120581) (119891

119896120575120573

120572minus 119891G]

119896119862120573

120572])

minus119891minus2G119904

120583G119898

120572119892120574120583119862ℎ

119897119904119862ℎ119896119898 minus C

119896

119897

(93)

R 120574

120572 120573120582= 119877

120574

120572 120573120582minus 119872

120581

120582

120597119865120574

120572120573

120597V120581+

120575119905119872

120574

120572120573

120575119905119906120582+119865

120574

120582120583119872

120583

120572120573+119872

120574

120582120583119865120583

120572120573

minus119872120574

120582120583119872

120583

120572120573+120575119905119873

120574

120572120582

120575119905119906120573+ 119865

120574

120573120583119873

120583

120572120582

+ 119873120574

120573120583119865120583

120572120582+ 119873

120574

120573120583119873

120583

120572120582+ 119873

120574

120582120583119872

120583

120572120573

+ 119873120583

120572120573119872

120574

120582120583+ (119892

119903119904120575120574

120582119891119904minus 119891119892

119903119904G120581

119904119862120574

120582120581)

times (119892120572120573119891119903minus 119891G120583

119903119862120572120573120583) minusC

120573

120582

(94)

where

119872119894

119895119896= 119872

119903

119896119862119894

119895119903+119872

119903

119895119862119894

119896119903minus119872

119903

ℎ119892119894ℎ119862119895119896119903

119872120574

120572120573= 119872

120583

120573119862120574

120572120583+119872

120583

120572119862120574

120573120583minus119872

120583

] 119892120574]119862120572120573120583

119873120574

120572120573= 119891

minus1(119891

120573120575120574

120572+ 119891

120572120575120574

120573minus 119891

120582119892120574120582119892120572120573)

(95)

and C119894

119895denotes the interchange of indices 119894 119895 and subtraction

ByTheorem 18 we have the following

Theorem 16 Let (1198721times119891119872

2 119865) be a flat twisted product

Finsler manifold and let (1198721 119865

1) be Riemannian If 119891 is a

function on1198722 only then (119872

1 119865

1) is locally flat

Similarly we get the following

Theorem 17 Let (1198721times119891119872

2 119865) be a flat twisted product

Finsler manifold and let (1198722 119865

2) be Riemannian If 119891 is a

function on 1198721 only then (119872

2 119865

2) is a space of positive

constant curvature ||119892119903119886119889119891||2

ISRN Geometry 9

Proof Since 1198722is Riemannain and 119891 is a function on 119872

1

then by (94) we obtain

R 120574

120572 120573120582= 119877

120574

120572 120573120582+1003817100381710038171003817119892119903119886119889119891

1003817100381710038171003817

2

(120575120574

120582119892120572120573minus 120575

120574

120573119892120572120582) (96)

Since (1198721times119891119872

2 119865) is flat then R120574

120572 120573120582= 0 Thus the proof is

complete

Theorem 18 Let (1198721times119891119872

2 119865) be a twisted product Rieman-

nian manifold and let 119891 be a function on 1198722 only Then

(1198721times119891119872

2 119865) is flat if and only if (119872

1 119865

1) is flat and the

Riemannian curvature of (1198722 119865

2) satisfies in the following

equation

119877120574

120572 120573120582= 120575119905119873

120574

120572120582

120575119905119906120573+ 119865

120574

120573120583119873

120583

120572120582+ 119873

120574

120573120583119865120583

120572120582+ 119873

120574

120573120583119873

120583

120572120582 minus C

120573

120582

(97)

5 Twisted Product Finsler Manifolds withNon-Riemannian Curvature Properties

There are several important non-Riemannian quantities suchas the Berwald curvature B the mean Berwald curvatureE and the Landsberg curvature L [22] They all vanishfor Riemannian metrics hence they are said to be non-Riemannian In this section we find some necessary and suf-ficient conditions underwhich a twisted product Riemannianmanifold are Berwaldian of isotropic Berwald curvatureof isotropic mean Berwald curvature First we prove thefollowing

Lemma 19 Let (1198721times119891119872

2 119865) be a twisted product Finsler

manifold Then the coefficients of Berwald curvature are asfollows

B120574

120572120573120582= 119861

120574

120572120573120582+ 119891

minus1(119862

120574]120582120572120573119891]119865

2

2+ 2119862

120574]120572120573119891]V120582

+ 2119862120574]120572120582119891]V120573 + 2119862

120574]120572119891]119892120582120573

+ 2119862120574]120582120573119891]V120572 + 2119862

120574]120573119891]119892120582120572

+ 2119862120574]120582119891]119892120572120573 minus 2119862120572120573120582119891

120574)

(98)

B119896

119894119895119897= 119861

119896

119894119895119897+ 119891119862

119896ℎ

119897119895119894119891ℎ1198652

2 (99)

B119896

119894120573119897= 2119891119862

119896ℎ

119894119897119891ℎV120573 (100)

B119896

120572120573119897= 2119891119892

120572120573119862119896ℎ

119897119891ℎ (101)

B119896

120572120573120582= minus 2119891119862

120572120573120582119891119896 (102)

B120574

119894120573120582= B120574

119894119895120582= B120574

119894119895119896= 0 (103)

Let (1198721times119891119872

2 119865) is a Berwald manifold Then we have

B119889

119886119887119888= 0 By using (102) we get

119862120572120573120582119891119896= 0 (104)

Multiplying this equation in 119892119896119903 we obtain

119862120572120573120582119891119903= 0 (105)

Thus if 119891 is not constant on1198721 then we have 119862

120572120573120582= 0 Also

from (101) we result that

119862119896ℎ

119897119891ℎ= 0 (106)

Differentiating this equation with respect to 119910119895 gives us

119862119896ℎ

119897119895119891ℎ= 0 (107)

Similarly we obtain

119862119896ℎ

119897119895119894119891ℎ= 0 (108)

Setting the last equation in (99) implies that 119861119896119894119895119897= 0 that

is (1198721 119865

1) is Berwaldian These explanations give us the

following theorem

Theorem 20 Let (1198721times119891119872

2 119865) be a twisted product Finsler

manifold and let119891 be not constant on1198721Then (119872

1times119891119872

2 119865)

is Berwaldian if and only if (1198721 119865

1) is Berwaldian (119872

2 119865

2) is

Riemannian and the equation 119862119896ℎ119897119891ℎ= 0 is hold

But if 119891 is constant on1198721 that is 119891

119894= 0 then we get the

following

Theorem 21 Let (1198721times119891119872

2 119865) be a twisted product Finsler

manifold and 119891 is constant on 1198721 Then (119872

1times119891119872

2 119865)

is Berwaldian if and only if (1198721 119865

1) is Berwaldian and

the Berwald curvature of (1198722 119865

2) satisfies in the following

equation

119861120574

120572120573120582= minus 119891

minus1(119862

120574]120573120572120582119891]119865

2

2+ 2119862

120574]120573120572119891]V120582 + 2119862

120574]120582120572119891]V120573

+ 2119862120574]120572119891]119892120582120573 + 2119862

120574]120573120582119891]V120572

+ 2119862120574]120573119891]119892120582120572 + 2119862

120574]120582119891]119892120572120573

minus 2119892120574]119862120572120573120582119891])

(109)

Here we consider twisted product Finsler manifold(119872

1times119891119872

2 119865) of isotropic Berwald curvature

Theorem 22 Every isotropic Berwald twisted product Finslermanifold (119872

1times119891119872

2 119865) is a Berwald manifold

Proof Let (1198721times119891119872

2 119865) be an isotropic Berwald manifold

Then we have

B119889

119886119887119888= 119888119865

minus1h119889

119886h119887119888+ h119889

119887h119886119888+ h119889

119888h119886119887+ 2C

119886119887119888y119889 (110)

where 119888 = 119888(x) is a function on119872 Setting 119886 = 119895 119887 = 119896 119888 = 119897and 119889 = 120574 and using (103) imply that

119888119865minus13

1198652119910119895119910119896119910119897V120574 minus V120574 (119910

119895119892119896119897+ 119910

119896119892119895119897+ 119910

119897119892119895119896) = 0 (111)

Multiplying the previous equation in 119910119895119910119896 we derive that119888119891

21198652

11198652

2= 0 Thus we have 119888 = 0 that is (119872

1times119891119872

2) is

Berwaldian

10 ISRN Geometry

Now we are going to study twisted product Finslermanifold of isotropicmean Berwald curvature For this workwe must compute the coefficients of mean Berwald curvatureof a twisted product Finsler manifold

Lemma 23 Let (1198721times119891119872

2 119865) be a twisted product Finsler

manifold Then the coefficients of mean Berwald curvature areas follows

E120572120573= 119864

120572120573+ 119891119892

120572120573119868ℎ119891ℎ+1

2119891119868

]120572120573119891]119865

2

2+ 119891

minus1119891]

times (119862120574]120572120573

V120574+ 119868

]120572V120573+ 119868

]120573V120572+ 119862

]120572120573+ 119868

]119892120572120573)

(112)

E119894119895= 119864

119894119895+1

2119891119868

ℎ

119895119894119891ℎ1198652

2 (113)

E119894120573= 119891119868

ℎ

119894119891ℎV120573 (114)

where 119864119894119895

and 119864120572120573

are the coefficients of mean Berwaldcurvature of (119872

1 119865

1) and (119872

2 119865

2) respectively

Proof By definition and Lemma 19 we get the proof

Theorem24 The twisted product Finslermanifold (1198721times119891119872

2

119865) is weakly Berwald if and only if (1198721 119865

1) is weakly Berwald

119868ℎ119891ℎ= 0 and the following hold

119864120572120573= minus

1

2119891119868

]120572120573119891]119865

2

2minus 119891

minus1119891]

times (119862120574]120572120573

V120574+ 119868

]120572V120573+ 119868

]120573V120572+ 119862

]120572120573+ 119868

]119892120572120573)

(115)

Proof If (1198721times119891119872

2) be a weakly Berwald manifold then we

have

E120572120573= E

119894119895= E

119894120573= 0 (116)

Thus by using (114) we result that 119868ℎ119894119891ℎ= 0 This equation

implies that

119868ℎ

119895119894119891ℎ= 0 119868

ℎ119891ℎ= 0 (117)

By setting these equations in (112) and (113) we conclude that119864119894119895= 0 and 119864

120572120573satisfies in (115)

Now if 119891 is constant on1198722 then (115) implies that 119864

120572120573=

0 Thus we conclude the following

Corollary 25 Let (1198721times119891119872

2 119865) be a twisted product Finsler

manifold and let 119891 be a function on 1198721 only Then

(1198721times119891119872

2 119865) is weakly Berwald if and only if (119872

1 119865

1) and

(1198722 119865

2) are weakly Berwald manifolds and 119868ℎ119891

ℎ= 0

Now we consider twisted product Finsler manifolds withisotropic mean Berwald curvature It is remarkable that as aconsequence of Lemma 23 we have the following

Lemma26 Twisted product Finslermanifold (1198721times119891119872

2 119865) is

isotropic mean Berwald manifold if and only if

119864120572120573+ 119891119892

120572120573119868ℎ119891ℎ+1

2119891119868

]120572120573119891]119865

2

2+ 119891

minus1119891]

times (119862120574]120572120573

V120574+ 119868

]120572V120573+ 119868

]120573V120572+ 119862

]120572120573+ 119868

]119892120572120573)

minus119899 + 1

2119888119891

2119865minus1(119892

120572120573minus1198912

1198652V120572V120573) = 0

(118)

119864119894119895+1

2119891119868

ℎ

119895119894119891ℎ1198652

2minus119899 + 1

2119888119865

minus1(119892

119894119895minus1

1198652119910119894119910119895) = 0 (119)

119888 (119899 + 1) 119865minus3119910119894+ 119891119868

ℎ

119894119891ℎ= 0 (120)

where 119888 = 119888(x) is a scalar function on119872

Theorem 27 Every twisted product Finsler manifold(119872

1times119891119872

2 119865) with isotropic mean Berwald curvature is a

weakly Berwald manifold

Proof Suppose that 119865 is isotropic mean Berwald twistedproduct Finsler metric Then differentiating (120) withrespect to V120574 gives us

119888 (119899 + 1) 1198912119865minus5V

120574119910119894= 0 (121)

Thus we conclude that 119888 = 0 This implies that 119865 reduces to aweakly Berwald metric

6 Locally Dually Flat Twisted ProductFinsler Manifolds

In [23] Amari and Nagaoka introduced the notion of duallyflat Riemannian metrics when they study the informationgeometry on Riemannian manifolds Information geometryhas emerged from investigating the geometrical structure ofa family of probability distributions and has been appliedsuccessfully to various areas including statistical inferencecontrol system theory andmultiterminal information theoryIn Finsler geometry Shen extends the notion of locally duallyflatness for Finsler metrics [17] Dually flat Finsler metricsform a special and valuable class of Finsler metrics in Finslerinformation geometry which play a very important role instudying flat Finsler information structure [24 25]

In this section we study locally dually flat twisted productFinsler metrics It is remarkable that a Finsler metric 119865 =119865(x y) on a manifold119872 is said to be locally dually flat if atany point there is a standard coordinate system (x119886 y119886) in119879119872such that it satisfies

12059721198652

120597x119887120597y119886y119887 = 2120597119865

2

120597x119886 (122)

In this case the coordinate (x119886) is called an adapted localcoordinate system Byusing (122) we can obtain the followinglemma

ISRN Geometry 11

Lemma 28 Let (1198721times119891119872

2 119865) be a twisted product Finsler

manifold Then 119865 is locally dually flat if and only if 1198651and 119865

2

satisfy in the following equations

12059721198652

1

120597119909119896120597119910119897119910119896= 2120597119865

2

1

120597119909119897+ 4119891119891

1198971198652

2 (123)

4119891119896V120573119910119896+ 119891

12059721198652

2

120597119906120572120597V120573V120572 + 4119891

120572V120573V120572 = 2119891

1205971198652

2

120597119906120573+ 4119891

1205731198652

2

(124)

Now let 119865 be a locally dually flat Finsler metric Takingderivative with respect to V120574 from (123) yields 119891

119897= 0 which

means that 119891 is a constant function on1198721 In this case the

relations (123) and (124) reduce to the following

12059721198652

1

120597119909119896120597119910119897119910119896= 2120597119865

2

1

120597119909119897 (125)

11989112059721198652

2

120597119906120572120597V120573V120572 + 4119891

120572V120573V120572 = 2119891

1205971198652

2

120597119906120573+ 4119891

1205731198652

2 (126)

By (125) we deduce that 1198651is locally dually flat

Now we assume that 1198651and 119865

2are locally dually flat

Finsler metrics Then we have

12059721198652

1

120597119909119896120597119910119897119910119896= 2120597119865

2

1

120597119909119897

12059721198652

2

120597119906120572120597V120573V120572 = 2

1205971198652

2

120597119906120573

(127)

By (127) we derive that (123) and (124) are hold if and only ifthe following hold

119891119897= 0 119891

120572V120573V120572 = 119891

1205731198652

2 (128)

Therefore we can conclude the following

Theorem 29 Let (1198721times119891119872

2 119865) be a twisted product Finsler

manifold

(i) If 119865 is locally dually flat then 1198651is locally dually flat 119891

is a function with respect to (119906120572) only and 1198652satisfies

in (126)(ii) If 119865

1and 119865

2are locally dually flat then 119865 is locally

dually flat if and only if 119891 is a function with respect(119906

120572) only and 119865

2satisfies in (128)

ByTheorem 29 we conclude the following

Corollary 30 There is not exist any locally dually flat propertwisted product Finsler manifold

References

[1] J K Beem P E Ehrlich and K L Easley Global LorentzianGeometry vol 202 of Monographs and Textbooks in Pure andApplied Mathematics Marcel Dekker New York NY USA 2ndedition 1996

[2] G Ganchev and V Mihova ldquoRiemannian manifolds of quasi-constant sectional curvaturesrdquo Journal fur die Reine und Ange-wandte Mathematik vol 522 pp 119ndash141 2000

[3] B OrsquoNeill Semi-Riemannian Geometry With Applications toRelativity vol 103 of Pure and Applied Mathematics AcademicPress New York NY USA 1983

[4] M Lohnherr and H Reckziegel ldquoOn ruled real hypersurfacesin complex space formsrdquo Geometriae Dedicata vol 74 no 3pp 267ndash286 1999

[5] B-Y Chen F Dillen L Verstraelen and L VranckenldquoLagrangian isometric immersions of a real-space-form119872119899

(119888)

into a complex-space-form119872119899(4119888)rdquoMathematical Proceedings

of the Cambridge Philosophical Society vol 124 no 1 pp 107ndash125 1998

[6] N Koike ldquoThe decomposition of curvature netted hypersur-facesrdquo Geometriae Dedicata vol 54 no 1 pp 1ndash11 1995

[7] B-y Chen Geometry of Submanifolds and its ApplicationsScience University of Tokyo Tokyo Japan 1981

[8] R Ponge and H Reckziegel ldquoTwisted products in pseudo-Riemannian geometryrdquo Geometriae Dedicata vol 48 no 1 pp15ndash25 1993

[9] B-Y Chen ldquoTwisted product CR-submanifolds in Kaehlermanifoldsrdquo Tamsui Oxford Journal of Mathematical Sciencesvol 16 no 2 pp 105ndash121 2000

[10] L Kozma I R Peter and H Shimada ldquoOn the twisted productof Finsler manifoldsrdquo Reports on Mathematical Physics vol 57no 3 pp 375ndash383 2006

[11] M Matsumoto and C Shibata ldquoOn semi-C-reducibility T-tensor= 0 and S4-likeness of Finsler spacesrdquo Journal of Math-ematics of Kyoto University vol 19 no 2 pp 301ndash314 1979

[12] B Najafi Z Shen and A Tayebi ldquoFinsler metrics of scalar flagcurvature with special non-Riemannian curvature propertiesrdquoGeometriae Dedicata vol 131 pp 87ndash97 2008

[13] E Peyghan andAHeydari ldquoConformal vector fields on tangentbundle of a Riemannian manifoldrdquo Journal of MathematicalAnalysis and Applications vol 347 no 1 pp 136ndash142 2008

[14] Z Shen Differential Geometry of Spray and Finsler SpacesKluwer Academic Publishers Dordrecht The Netherlands2001

[15] M Matsumoto and S-I Hojo ldquoA conclusive theorem on C-reducible Finsler spacesrdquo The Tensor Society vol 32 no 2 pp225ndash230 1978

[16] D Bao S-S Chern and Z Shen An Introduction to Riemann-Finsler Geometry vol 200 of Graduate Texts in MathematicsSpringer New York NY USA 2000

[17] Z Shen ldquoRiemann-Finsler geometry with applications to infor-mation geometryrdquo Chinese Annals of Mathematics B vol 27 no1 pp 73ndash94 2006

[18] Y Ichijyo ldquoFinsler manifolds modeled on a Minkowski spacerdquoJournal of Mathematics of Kyoto University vol 16 no 3 pp639ndash652 1976

[19] X Chen and Z Shen ldquoOn Douglas metricsrdquo PublicationesMathematicae Debrecen vol 66 no 3-4 pp 503ndash512 2005

[20] L Berwald ldquoUber Parallelubertragung in Raumen mit allge-meiner Massbestimmungrdquo Jahresbericht der DMVmdashDeutscheMathematiker-Vereinigung vol 34 pp 213ndash220 1926

[21] M Matsumoto ldquoOn Finsler spaces with Randersrsquo metric andspecial forms of important tensorsrdquo Journal of Mathematics ofKyoto University vol 14 pp 477ndash498 1974

12 ISRN Geometry

[22] A Tayebi and E Peyghan ldquoOn special Berwald metricsrdquoSymmetry Integrability and Geometry vol 6 article 008 9pages 2010

[23] S-I Amari and H NagaokaMethods of Information Geometryvol 191 of Translations of Mathematical Monographs AmericanMathematical Society Providence RI USA 2000

[24] X Cheng Z Shen and Y Zhou ldquoOn locally dually flat Finslermetricsrdquo International Journal ofMathematics vol 21 no 11 pp1531ndash1543 2010

[25] Q Xia ldquoOn a class of locally dually flat Finsler metrics ofisotropic flag curvaturerdquo Publicationes Mathematicae Debrecenvol 78 no 1 pp 169ndash190 2011

Submit your manuscripts athttpwwwhindawicom

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Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

ISRN Geometry 5

Definition 5 Using decomposition (33) the twisted almosttangent structure 119869119905 119867119879119872∘

rarr 119881119879119872∘ is defined by

119869119905120597

120597119910119894otimes 119889119909

119894+120597

120597V120572otimes 119889119906

120572 (43)

or

119869119905(120575119905

120575119905119909119894) =

120597

120597119910119894 119869

119905(120575119905

120575119905119906120572) =

120597

120597V120572

119869119905(120597

120597119910119894) = 119869

119905(120597

120597V120572) = 0

(44)

Thus we result that

(119869119905)2

= 0 ker 119869119905 = 119868119898119869119905 = 119881119879119872∘ (45)

Here we introduce some geometrical objects of twistedproduct Finsler manifold In order to simplify the equationswe rewritten the basis of119867119879119872∘ and 119881119879119872∘ as follows

120575119905

120575119905x119886=120575119905

120575119905119909119894120575119894

119886+120575119905

120575119905119906120572120575120572

119886

120597

120597y119886=120597

120597119910119894120575119894

119886+120597

120597V120572120575120572

119886

(46)

Thus

119879119879119872∘= span 120575

119905

120575119905x119886120597

120597y119886 (47)

The Lie brackets of this basis is given by

[120575119905

120575119905x119886120575119905

120575119905x119887] = R119888

119886119887

120597

120597y119888

[120575119905

120575119905x119886120597

120597y119887] = G119888

119886119887

120597

120597y119888

[120597

120597y119886120597

120597y119887] = 0

(48)

where

(i) R119888

119886119887=120575119905G119888

119886

120575119905x119887minus120575119905G119888

119887

120575119905x119886 (49)

(ii) G119888

119886119887=120597G119888

119886

120597y119887 (50)

Therefore we have the following

Corollary 6 Let (1198721times119891119872

2 119865) be a twisted product Finsler

manifold Then

R119888

119886119887= (R119896

119894119895R119896

119894120573R119896

120572119895R119896

120572120573R120574

119894119895R120574

119894120573R120574

120572119895R120574

120572120573)

(51)

where

R119896

119894119895=120575119905G119896

119894

120575119905119909119895minus

120575119905G119896

119895

120575119905119909119894 R119896

119894120573=120575119905G119896

119894

120575119905119906120573minus

120575119905G119896

120573

120575119905119909119894

R119896

120572119895=120575119905G119896

120572

120575119905119909119895minus

120575119905G119896

119895

120575119905119906120572 R119896

120572120573=120575119905G119896

120572

120575119905119906120573minus

120575119905G119896

120573

120575119905119906120572

R120574

119894119895=120575119905G120574

119894

120575119905119909119895minus

120575119905G120574

119895

120575119905119909119894 R120574

119894120573=120575119905G120574

119894

120575119905119906120573minus

120575119905G120574

120573

120575119905119909119894

R120574

120572119895=120575119905G120574

120572

120575119905119909119895minus

120575119905G120574

119895

120575119905119906120572 R120574

120572120573=120575119905G120574

120572

120575119905119906120573minus

120575119905G120574

120573

120575119905119906120572

(52)

With a simple calculation we have the following

Corollary 7 Let (1198721times119891119872

2 119865) be a twisted product Finsler

manifold Then

G119888

119886119887= (G119896

119894119895G119896

119894120573G119896

120572119895G119896

120572120573G120574

119894119895G120574

119894120573G120574

120572119895G120574

120572120573) (53)

where

G120574

120572120573=120597G120574

120572

120597V120573

= 119866120574

120572120573+ 119891

minus1(119862

120574120582

1205721205731198911205821198652

2+ 2119862

120574120582

120572119891120582V120573+ 2119862

120574120582

120573119891120582V120572

minus 119891120574119892120572120573+ 119891

120573120575120574

120572+ 119891

120572120575120574

120573) = G120574

120573120572

G119896

119894119895=120597G119896

119894

120597119910119895= 119866

119896

119894119895+ 119862

119896ℎ

119894119895119891119891

ℎ1198652

2= G119896

119895119894

G119896

119894120573=120597G119896

119894

120597V120573= 2119862

119896ℎ

119894119891119891

ℎV120573= G119896

120573119894

G119896

120572120573=120597G119896

120572

120597V120573= minus119891119891

119896119892120572120573= G119896

120573120572

G120574

119894120573=120597G120574

119894

120597V120573= 119891

minus1119891119894120575120574

120573= G120574

120573119894

G120574

119894119895=120597G120574

119894

120597119910119895= G120574

119895119894= 0

(54)

where 119862119896ℎ119894119895= 120597119862

119896ℎ

119894120597119910

119895 Apart from G119888

119886119887 the functions F119888

119886119887are

given by

F119888119886119887=1

2g119888119890 (120575

119905g119890119886

120575119905x119887+120575119905g

119890119887

120575119905x119886minus120575119905g

119886119887

120575119905x119890) (55)

Corollary 8 Let (1198721times119891119872

2 119865) be a twisted product Finsler

manifold Then

F119888119886119887= (F119896

119894119895 F119896

119894120573 F119896

120572119895 F119896

120572120573 F120574

119894119895 F120574

119894120573 F120574

120572119895 F120574

120572120573) (56)

6 ISRN Geometry

where

F119896119894119895= 119865

119896

119894119895minus (119872

119903

119895119862119896

119894119903+119872

119903

119894119862119896

119895119903minus119872

119903

ℎ119862119894119895119903119892119896ℎ) (57)

F119896119894120573= minusG119903

120573119862119896

119894119903= F119896

120573119894 (58)

F119896120572120573= minus119891119891

119896119892120572120573+ 119891

2119892119896ℎG120582

ℎ119862120572120573120582 (59)

F120574119894119895= 119891

minus2119892120574120582G119903

120582119862119894119895119903 (60)

F120574119894120573= 119891

minus1119891119894120575120574

120573minus G120572

119894119862120574

120572120573= F120574

120573119894 (61)

F120574120572120573= 119865

120574

120572120573+ 119873

120574

120572120573minus (119872

120583

120573119862120574

120572120583+119872

120583

120572119862120574

120573120583minus119872

120583

120582119862120572120573120583119892120574120582)

(62)

119865119896

119894119895=1

2119892119896ℎ(120575119892

ℎ119894

120575119909119895+

120575119892ℎ119895

120575119909119894minus

120575119892119894119895

120575119909ℎ)

119865120574

120572120573=1

2119892120574120582(120575119892

120582120572

120575119906120573+

120575119892120582120573

120575119906120572minus

120575119892120572120573

120575119906120582)

119872119903

119894= 119862

119903ℎ

119894119891119891

ℎ1198652

2

119872120583

120572= 119891

minus1(119862

120583120574

1205721198911205741198652

2+ 119891

119903119910119903120575120583

120572+ 119891

120574V120574120575120583

120572minus 119892

120583120574119891120574V120572+ 119891

120572V120583)

119873120574

120572120573= 119891

minus1(119891

120573120575120574

120572+ 119891

120572120575120574

120573minus 119891

120582119892120574120582119892120572120573)

(63)

Proof By using (55) we have

F119896119894119895=1

2119892119896ℎ(120575119905119892ℎ119894

120575119905119909119895+

120575119905119892ℎ119895

120575119905119909119894minus

120575119905119892119894119895

120575119905119909ℎ) (64)

Since 119892119894119895is a function with respect to (119909 119910) then by (25) and

(30) we obtain

120575119905119892ℎ119894

120575119905119909119895=120575119892

ℎ119894

120575119909119895minus 2119872

119903

119895119862ℎ119894119903 (65)

Interchanging 119894 119895 and ℎ in the previous equation gives us

120575119905119892ℎ119895

120575119905119909119894=

120575119892ℎ119895

120575119909119894minus 2119872

119903

119894119862ℎ119895119903

120575119905119892119894119895

120575119905119909ℎ=

120575119892119894119895

120575119909ℎminus 2119872

119903

ℎ119862119894119895119903

(66)

Putting these equation in (64) give us (57) In the similar waywe can prove the another relation

By using (i) of (23) and (57)ndash(62) we can conclude thefollowing

Lemma 9 Let (1198721times119891119872

2 119865) be a twisted product Finsler

manifold Then y119888F119886119887119888= G119886

119887 where F119886

119887119888and G119886

119887are defined by

(55) and (i) of (23) respectively

The Cartan torsion is one of the most important non-Riemannian quantity in Finsler geometry and it is first

introduced by Finsler and emphasized by Cartan whichmeasures a departure from a Riemannian manifold Moreprecisely a Finsler metric reduces to a Riemannian metricif and only if it has vanishing Cartan torsion The localcomponents of Cartan tensor field of the twisted Finslermanifold (119872

1times119891119872

2 119865) is defined by

C119886

119887119888=1

2g119886119890 120597g119887119890120597y119888 (67)

From this definition we conclude the following

Lemma 10 Let119862119896119894119895and119862120574

120572120573be the local components of Cartan

tensor field on1198721and119872

2 respectively Then one has

C119888

119886119887= (C119896

119894119895C119896

119894120573C119896

120572119895C119896

120572120573C120574

119894119895C120574

119894120573C120574

120572119895C120574

120572120573) (68)

where

C119896

119894119895=1

2119892119896ℎ120597119892

119894119895

120597119910ℎ= 119862

119896

119894119895

C120574

120572120573=1

2119892120574120582120597119892

120572120573

120597V120582= 119862

120574

120572120573

(69)

and C119896

119894120573= C119896

120572119895= C119896

120572120573= C120574

119894119895= C120574

119894120573= C120574

120572119895= 0

By using the Lemma 10 we can get the following

Corollary 11 Let (1198721times119891119872

2 119865) be a twisted product Finsler

manifoldThen (1198721times119891119872

2 119865) is a Riemannianmanifold if and

only if (1198721 119865

1) and (119872

2 119865

2) are Riemannian manifold

Various interesting special forms of Cartan tensors havebeen obtained by some Finslerians [11] The Finsler spaceshaving such special forms have been called C-reducible C2-like semi-C-reducible and so forth In [21] Matsumotointroduced the notion of C-reducible Finsler metrics andproved that any Randers metric is C-reducible Later onMatsumoto-Hojo proves that the converse is true too [15]

Here we define the Matsumoto twisted tensorM119886119887119888

for atwisted product Finsler manifold (119872

1times119891119872

2 119865) as follows

M119886119887119888= C

119886119887119888minus1

119899 + 1I

119886h119887119888+ I

119887h119886119888+ I

119888h119886119887 (70)

where I119886= g119887119888C

119886119887119888C

119886119887119888= g

119888119889C119889

119886119887 andh

119886119887= g

119886119887minus(1119865

2)y

119886y119887

By attention to the previous equation and relations

C119894119895119896= 119862

119894119895119896 C

120572120573120574= 119891

2119862120572120573120574 (71)

we obtain

M120572119895119896= minus

1

119899 + 1119868

120572(119892

119895119896minus1

1198652119910119895119910119896) minus1198912

1198652V120572(119868

119895119910119896+ 119868

119896119910119895)

(72)

Contracting the previous equation in 119910119895119910119896 gives us

119910119895119910119896M

120572119895119896= minus11989121198652

11198652

2

(119899 + 1) 1198652119868120572 (73)

ISRN Geometry 7

Similarly we obtain

V120582V120573M119894120573120582= minus11989121198652

11198652

2

(119899 + 1) 1198652119868119894 (74)

Therefore if M119894120573120582= M

120572119895119896= 0 then we get 119868

119894= 119868

120572= 0 that

is (1198721 119865

1) and (119872

2 119865

2) are Riemannian manifolds Thus we

have the following

Theorem 12 There is not exist any C-reducible twisted prod-uct Finsler manifold

Now we are going to consider semi-C-reducible twistedproduct Finsler manifold (119872

1times119891119872

2 119865) Let (119872

1times119891119872

2 119865) be

a semi-C-reducible twisted product Finsler manifold Thenwe have

C119886119887119888=119901

119899 + 1I

119886h119887119888+ I

119887h119886119888+ I

119888h119886119887 +119902

C2I119886I119887I119888 (75)

where C2= I119886I

119886and 119901 and 119902 are scalar function on119872

1times119891119872

2

with 119901 + 119902 = 1 This equation gives us

0 = C120572119895119896

=119901

119899 + 1119868

120572(119892

119895119896minus1

1198652119910119895119910119896) minus1198912

1198652V120572(119868

119895119910119896+ 119868

119896119910119895)

+119902

C2119868120572119868119895119868119896

(76)

Contractiing the previous equation with 119910119895119910119896 implies that

11990111989121198652

11198652

2119868120572= 0 (77)

Therefore we have 119901 = 0 or 119868120572= 0 If 119901 = 0 then

119865 is 1198622-like metric But if 119901 = 0 then 119868120572= 0 that is

1198652is Riemannian metric In this case with similar way

we conclude that 1198651is Riemannian metric But definition

119865 cannot be a Riemannian metric Therefore we have thefollowing

Theorem 13 Every semi-C-reducible twisted product Finslermanifold (119872

1times119891119872

2 119865) is a 1198622-like manifold

4 Riemannian Curvature

The Riemannian curvature of twisted product Finsler man-ifold (119872

1times119891119872

2 119865) with respect to Berwald connection is

given by

R 119886

119887 119888119889=120575119905F119886

119887119888

120575119905x119889minus120575119905F119886

119887119889

120575119905x119888+ F119886

119889119890F119890119887119888minus F119886

119888119890F119890119887119889 (78)

Lemma 14 Let (1198721times119891119872

2 119865) be a twisted product Finsler

manifold Then one has

R119886

119888119889= y119887R 119886

119887 119888119889 (79)

where R119886

119888119889and y119887R 119886

119887 119888119889are given by (50) and (78)

Proof By using (78) we have

y119887R 119894

119887 119896119897= y119887120575119905F119894

119887119896

120575119905x119897minus y119887120575119905F119894

119887119897

120575119905x119896+ y119887F119894

119897119890F119890119887119896 minus y119887F119894

119896119890F119890119887119897 (80)

By using Corollary 8 and Lemma 9 we obtain

y119887120575119905F119894

119887119896

120575119905x119897=120575119905G119894

119896

120575119905119909119897+ F119894

119895119896G119895

119897+ F119894

120573119896G120573

119897

y119887F119894119897119890F119890119887119896= F119894

119897ℎGℎ

119896+ F119894

119897120574G120574

119896

(81)

Interchanging 119894 and 119895 in the previous equation implies that

y119887120575119905F119894

119887119897

120575119905x119896=120575119905G119894

119897

120575119905119909119896+ F119894

119895119897G119895

119896+ F119894

120573119897G120573

119896

y119887F119894119896119890F119890119887119897= F119894

119896ℎGℎ

119897+ F119894

119896120574G120574

119897

(82)

Setting (81) and (82) in (80) gives us y119887R119894

119887 119896119897= R119894

119896119897 In the

similar way we can obtain this relation for another indices

Using (78) we can compute the Riemannian curvature ofa twisted product Finsler manifold

Lemma 15 Let (1198721times119891119872

2 119865) be a twisted product Finsler

manifoldThen the coefficients of Riemannian curvature are asfollows

R119894

119895 119896119897= 119877

119894

119895 119896119897

minus 119872119903

119897

120597119865119894

119895119896

120597119910119903+

120575119905119872

119894

119895119896

120575119905119909119897+119865

119894

119897ℎ119872

ℎ

119895119896+119872

119894

119897ℎ119865ℎ

119895119896minus119872

119894

119897ℎ119872

ℎ

119895119896

+ 119891minus2119892120572120574G119903

120572G119898

120574119862119894

119897119903119862119895119896119898 minusC

119896

119897

(83)

R 119894

120572 119896119897= minus

120575119905

120575119905119909119897(G119903

120572119862119894

119896119903) minus (119865

119894

119903119897minus119872

119894

119903119897)G119898

120572119862119903

119896119898

minus119891minus1G119903

120573119862119894

119897119903119891119896120575120573

120572+ G119903

120573G120583

119896119862119894

119897119903119862120573

120572120583 minus C

119896

119897

(84)

R 119894

119895 120573120582= minus

120575119905

120575119905119906120582(G119903

120573119862119894

119895119903) + G119898

120582G119897

120573119862119894

119903119898119862119903

119895119897

minus (119891119894119892120572120582minus 119891G120583

ℎ119892119894ℎ119862120572120582120583) (119891

119895120575120572

120573minus 119891G]

119895119862120572

120573])

minus C120573

120582

(85)

8 ISRN Geometry

R 119894

120572 120573119897=120575119905

120575119905119906120573(G119903

120572119862119894

119897119903) minus120575119905

120575119905119909119897119891 (119891

119894119892120572120573minus 119891G120582

ℎ119892119894ℎ119862120572120573120582)

minus G119898

120573G119904

120572119862119894

119903119898119862119903

119897119904+ (119891

119894119892120583120573minus 119891G120582

ℎ119892119894ℎ119862120583120573120582)

times (119891119897120575120583

120572minus 119891G]

119897119862120583

120572]) minus 119891119892119903ℎ(119865

119894

119903119897minus119872

119894

119903119897)

times (119891ℎ119892120572120573minus119891G120582

ℎ119862120572120573120582)minusG119903

120583119862119894

119897119903(119865

120583

120572120573+ 119873

120583

120572120573minus119872

120583

120572120573)

(86)

R 119894

119895 120573119897= minus

120575119905

120575119905119909119897(G119903

120573119862119894

119895119903) minus

120575119905

120575119905119906120573(119865

119894

119895119897minus119872

119894

119895119897)

minus (119865119894

119897119903minus119872

119894

119897119903)G119904

120573119862119903

119895119904minus 119891

minus1G119903

120572119862119894

119897119903

times (119891119895120575120572

120573minus 119891G120583

119895119862120572

120573120583) + G119904

120573119862119894

119903119904(119865

119903

119895119897minus119872

119903

119895119897)

+ 119891minus1G119903

120583119862119895119897119903(119891

119894120575120583

120573minus 119891G120582

ℎ119892119894ℎ119862120583

120573120582)

(87)

R 119894

120572 120573120582= minus

120575119905

120575119905119906120582(119891119891

119894119892120572120573minus 119891

2119892119894ℎG120583

ℎ119862120572120573120583) + 119891G119904

120582119862119894

119903119904

times(119891119903119892120572120573minus119891G120583

119897119862120572120573120583119892119903119897)minus119891 (119891

119894119892120582120583minus119891119892

119894ℎG120581

ℎ119862120582120583120581)

times (119865120583

120572120573+ 119873

120583

120572120573minus119872

120583

120572120573) minus C

120573

120582

(88)

R 120574

119895 119896119897=

120575119905

120575119905119909119897(119891

minus2119892120574120582G119903

120582119862119895119896119903) + 119891

minus2119892120574120582G119904

120582119862119897119903119904

times (119865119903

119895119896minus119872

119903

119895119896) + 119891

minus3G119903

120583119862119895119896119903(119891

119897119892120574120583minus 119891G120572

119897119862120574120583

120572)

minus C119896

119897

(89)

R 120574

119895 120573119897=120575119905

120575119905119909119897(119891

minus1119891119895120575120574

120573minus 119891G120572

119895119862120574

120572120573)

minus120575119905

120575119905119906120573(119891

minus2119892120574120582G119903

120582119862119895119897119903) minus 119891

minus2119892120574120582G119904

120582G119898

120573119862ℎ

119897119904119862ℎ119895119898

+ 119891minus2(119891

119897120575120574

120583minus 119891G120572

119897119862120574

120583120572) (119891

119895120575120583

120573minus 119891G]

119895119862120583

120573])

minus 119891minus1(119891

119903120575120574

120573minus 119891G120572

119903119862120574

120573120572) (119865

119903

119895119897minus119872

119903

119895119897)

minus 119891minus2119892120583120582G119903

120582119862119895119897119903(119865

120574

120573120583+ 119873

120574

120573120583minus119872

120574

120573120583)

(90)

R 120574

120572 120573119897=120575119905

120575119905119909119897(119865

120574

120572120573+ 119873

120574

120572120573minus119872

120574

120572120573)

minus120575119905

120575119905119906120573(119891

minus1119891119897120575120574

120572minus G120583

119897119862120574

120572120583)

minus 119891minus1119892120574120582G119904

120582119862ℎ

119897119904(119891

ℎ119892120572120573minus 119891G120583

ℎ119862120572120573120583)

+ 119891minus1(119891

119897120575120574

120583minus 119891G120581

119897119862120574

120583120581) (119865

120583

120572120573+ 119873

120583

120572120573minus119872

120583

120572120573)

+ 119891minus1G119904

120572119862119903

119897119904(119891

119903120575120574

120573minus 119891G120581

119903119862120574

120573120581)

minus 119891minus1(119891

119897120575120583

120572minus 119891G120581

119897119862120583

120572120581) (119865

120574

120573120583+ 119873

120574

120573120583minus119872

120574

120573120583)

(91)

R 120574

119895 120573120582=

120575119905

120575119905119906120582(119891

minus1119891119895120575120574

120573minus G120572

119895119862120574

120572120573)

+ 119891minus1(119865

120574

120572120582+ 119873

120574

120572120582minus119872

120574

120572120582) (119891

119895120575120572

120573minus 119891G]

119895119862120572

120573])

minus119891minus1G119898

120573119862119903

119895119898(119891

119903120575120574

120582minus 119891G120572

119903119862120574

120582120572) minus C

120573

120582

(92)

R 120574

120572 119896119897=

120575119905

120575119905119909119897(119891

minus1119891119896120575120574

120572minus G120583

119896119862120574

120572120583)

+ 119891minus2(119891

119897120575120574

120573minus 119891G120581

119897119862120574

120573120581) (119891

119896120575120573

120572minus 119891G]

119896119862120573

120572])

minus119891minus2G119904

120583G119898

120572119892120574120583119862ℎ

119897119904119862ℎ119896119898 minus C

119896

119897

(93)

R 120574

120572 120573120582= 119877

120574

120572 120573120582minus 119872

120581

120582

120597119865120574

120572120573

120597V120581+

120575119905119872

120574

120572120573

120575119905119906120582+119865

120574

120582120583119872

120583

120572120573+119872

120574

120582120583119865120583

120572120573

minus119872120574

120582120583119872

120583

120572120573+120575119905119873

120574

120572120582

120575119905119906120573+ 119865

120574

120573120583119873

120583

120572120582

+ 119873120574

120573120583119865120583

120572120582+ 119873

120574

120573120583119873

120583

120572120582+ 119873

120574

120582120583119872

120583

120572120573

+ 119873120583

120572120573119872

120574

120582120583+ (119892

119903119904120575120574

120582119891119904minus 119891119892

119903119904G120581

119904119862120574

120582120581)

times (119892120572120573119891119903minus 119891G120583

119903119862120572120573120583) minusC

120573

120582

(94)

where

119872119894

119895119896= 119872

119903

119896119862119894

119895119903+119872

119903

119895119862119894

119896119903minus119872

119903

ℎ119892119894ℎ119862119895119896119903

119872120574

120572120573= 119872

120583

120573119862120574

120572120583+119872

120583

120572119862120574

120573120583minus119872

120583

] 119892120574]119862120572120573120583

119873120574

120572120573= 119891

minus1(119891

120573120575120574

120572+ 119891

120572120575120574

120573minus 119891

120582119892120574120582119892120572120573)

(95)

and C119894

119895denotes the interchange of indices 119894 119895 and subtraction

ByTheorem 18 we have the following

Theorem 16 Let (1198721times119891119872

2 119865) be a flat twisted product

Finsler manifold and let (1198721 119865

1) be Riemannian If 119891 is a

function on1198722 only then (119872

1 119865

1) is locally flat

Similarly we get the following

Theorem 17 Let (1198721times119891119872

2 119865) be a flat twisted product

Finsler manifold and let (1198722 119865

2) be Riemannian If 119891 is a

function on 1198721 only then (119872

2 119865

2) is a space of positive

constant curvature ||119892119903119886119889119891||2

ISRN Geometry 9

Proof Since 1198722is Riemannain and 119891 is a function on 119872

1

then by (94) we obtain

R 120574

120572 120573120582= 119877

120574

120572 120573120582+1003817100381710038171003817119892119903119886119889119891

1003817100381710038171003817

2

(120575120574

120582119892120572120573minus 120575

120574

120573119892120572120582) (96)

Since (1198721times119891119872

2 119865) is flat then R120574

120572 120573120582= 0 Thus the proof is

complete

Theorem 18 Let (1198721times119891119872

2 119865) be a twisted product Rieman-

nian manifold and let 119891 be a function on 1198722 only Then

(1198721times119891119872

2 119865) is flat if and only if (119872

1 119865

1) is flat and the

Riemannian curvature of (1198722 119865

2) satisfies in the following

equation

119877120574

120572 120573120582= 120575119905119873

120574

120572120582

120575119905119906120573+ 119865

120574

120573120583119873

120583

120572120582+ 119873

120574

120573120583119865120583

120572120582+ 119873

120574

120573120583119873

120583

120572120582 minus C

120573

120582

(97)

5 Twisted Product Finsler Manifolds withNon-Riemannian Curvature Properties

There are several important non-Riemannian quantities suchas the Berwald curvature B the mean Berwald curvatureE and the Landsberg curvature L [22] They all vanishfor Riemannian metrics hence they are said to be non-Riemannian In this section we find some necessary and suf-ficient conditions underwhich a twisted product Riemannianmanifold are Berwaldian of isotropic Berwald curvatureof isotropic mean Berwald curvature First we prove thefollowing

Lemma 19 Let (1198721times119891119872

2 119865) be a twisted product Finsler

manifold Then the coefficients of Berwald curvature are asfollows

B120574

120572120573120582= 119861

120574

120572120573120582+ 119891

minus1(119862

120574]120582120572120573119891]119865

2

2+ 2119862

120574]120572120573119891]V120582

+ 2119862120574]120572120582119891]V120573 + 2119862

120574]120572119891]119892120582120573

+ 2119862120574]120582120573119891]V120572 + 2119862

120574]120573119891]119892120582120572

+ 2119862120574]120582119891]119892120572120573 minus 2119862120572120573120582119891

120574)

(98)

B119896

119894119895119897= 119861

119896

119894119895119897+ 119891119862

119896ℎ

119897119895119894119891ℎ1198652

2 (99)

B119896

119894120573119897= 2119891119862

119896ℎ

119894119897119891ℎV120573 (100)

B119896

120572120573119897= 2119891119892

120572120573119862119896ℎ

119897119891ℎ (101)

B119896

120572120573120582= minus 2119891119862

120572120573120582119891119896 (102)

B120574

119894120573120582= B120574

119894119895120582= B120574

119894119895119896= 0 (103)

Let (1198721times119891119872

2 119865) is a Berwald manifold Then we have

B119889

119886119887119888= 0 By using (102) we get

119862120572120573120582119891119896= 0 (104)

Multiplying this equation in 119892119896119903 we obtain

119862120572120573120582119891119903= 0 (105)

Thus if 119891 is not constant on1198721 then we have 119862

120572120573120582= 0 Also

from (101) we result that

119862119896ℎ

119897119891ℎ= 0 (106)

Differentiating this equation with respect to 119910119895 gives us

119862119896ℎ

119897119895119891ℎ= 0 (107)

Similarly we obtain

119862119896ℎ

119897119895119894119891ℎ= 0 (108)

Setting the last equation in (99) implies that 119861119896119894119895119897= 0 that

is (1198721 119865

1) is Berwaldian These explanations give us the

following theorem

Theorem 20 Let (1198721times119891119872

2 119865) be a twisted product Finsler

manifold and let119891 be not constant on1198721Then (119872

1times119891119872

2 119865)

is Berwaldian if and only if (1198721 119865

1) is Berwaldian (119872

2 119865

2) is

Riemannian and the equation 119862119896ℎ119897119891ℎ= 0 is hold

But if 119891 is constant on1198721 that is 119891

119894= 0 then we get the

following

Theorem 21 Let (1198721times119891119872

2 119865) be a twisted product Finsler

manifold and 119891 is constant on 1198721 Then (119872

1times119891119872

2 119865)

is Berwaldian if and only if (1198721 119865

1) is Berwaldian and

the Berwald curvature of (1198722 119865

2) satisfies in the following

equation

119861120574

120572120573120582= minus 119891

minus1(119862

120574]120573120572120582119891]119865

2

2+ 2119862

120574]120573120572119891]V120582 + 2119862

120574]120582120572119891]V120573

+ 2119862120574]120572119891]119892120582120573 + 2119862

120574]120573120582119891]V120572

+ 2119862120574]120573119891]119892120582120572 + 2119862

120574]120582119891]119892120572120573

minus 2119892120574]119862120572120573120582119891])

(109)

Here we consider twisted product Finsler manifold(119872

1times119891119872

2 119865) of isotropic Berwald curvature

Theorem 22 Every isotropic Berwald twisted product Finslermanifold (119872

1times119891119872

2 119865) is a Berwald manifold

Proof Let (1198721times119891119872

2 119865) be an isotropic Berwald manifold

Then we have

B119889

119886119887119888= 119888119865

minus1h119889

119886h119887119888+ h119889

119887h119886119888+ h119889

119888h119886119887+ 2C

119886119887119888y119889 (110)

where 119888 = 119888(x) is a function on119872 Setting 119886 = 119895 119887 = 119896 119888 = 119897and 119889 = 120574 and using (103) imply that

119888119865minus13

1198652119910119895119910119896119910119897V120574 minus V120574 (119910

119895119892119896119897+ 119910

119896119892119895119897+ 119910

119897119892119895119896) = 0 (111)

Multiplying the previous equation in 119910119895119910119896 we derive that119888119891

21198652

11198652

2= 0 Thus we have 119888 = 0 that is (119872

1times119891119872

2) is

Berwaldian

10 ISRN Geometry

Now we are going to study twisted product Finslermanifold of isotropicmean Berwald curvature For this workwe must compute the coefficients of mean Berwald curvatureof a twisted product Finsler manifold

Lemma 23 Let (1198721times119891119872

2 119865) be a twisted product Finsler

manifold Then the coefficients of mean Berwald curvature areas follows

E120572120573= 119864

120572120573+ 119891119892

120572120573119868ℎ119891ℎ+1

2119891119868

]120572120573119891]119865

2

2+ 119891

minus1119891]

times (119862120574]120572120573

V120574+ 119868

]120572V120573+ 119868

]120573V120572+ 119862

]120572120573+ 119868

]119892120572120573)

(112)

E119894119895= 119864

119894119895+1

2119891119868

ℎ

119895119894119891ℎ1198652

2 (113)

E119894120573= 119891119868

ℎ

119894119891ℎV120573 (114)

where 119864119894119895

and 119864120572120573

are the coefficients of mean Berwaldcurvature of (119872

1 119865

1) and (119872

2 119865

2) respectively

Proof By definition and Lemma 19 we get the proof

Theorem24 The twisted product Finslermanifold (1198721times119891119872

2

119865) is weakly Berwald if and only if (1198721 119865

1) is weakly Berwald

119868ℎ119891ℎ= 0 and the following hold

119864120572120573= minus

1

2119891119868

]120572120573119891]119865

2

2minus 119891

minus1119891]

times (119862120574]120572120573

V120574+ 119868

]120572V120573+ 119868

]120573V120572+ 119862

]120572120573+ 119868

]119892120572120573)

(115)

Proof If (1198721times119891119872

2) be a weakly Berwald manifold then we

have

E120572120573= E

119894119895= E

119894120573= 0 (116)

Thus by using (114) we result that 119868ℎ119894119891ℎ= 0 This equation

implies that

119868ℎ

119895119894119891ℎ= 0 119868

ℎ119891ℎ= 0 (117)

By setting these equations in (112) and (113) we conclude that119864119894119895= 0 and 119864

120572120573satisfies in (115)

Now if 119891 is constant on1198722 then (115) implies that 119864

120572120573=

0 Thus we conclude the following

Corollary 25 Let (1198721times119891119872

2 119865) be a twisted product Finsler

manifold and let 119891 be a function on 1198721 only Then

(1198721times119891119872

2 119865) is weakly Berwald if and only if (119872

1 119865

1) and

(1198722 119865

2) are weakly Berwald manifolds and 119868ℎ119891

ℎ= 0

Now we consider twisted product Finsler manifolds withisotropic mean Berwald curvature It is remarkable that as aconsequence of Lemma 23 we have the following

Lemma26 Twisted product Finslermanifold (1198721times119891119872

2 119865) is

isotropic mean Berwald manifold if and only if

119864120572120573+ 119891119892

120572120573119868ℎ119891ℎ+1

2119891119868

]120572120573119891]119865

2

2+ 119891

minus1119891]

times (119862120574]120572120573

V120574+ 119868

]120572V120573+ 119868

]120573V120572+ 119862

]120572120573+ 119868

]119892120572120573)

minus119899 + 1

2119888119891

2119865minus1(119892

120572120573minus1198912

1198652V120572V120573) = 0

(118)

119864119894119895+1

2119891119868

ℎ

119895119894119891ℎ1198652

2minus119899 + 1

2119888119865

minus1(119892

119894119895minus1

1198652119910119894119910119895) = 0 (119)

119888 (119899 + 1) 119865minus3119910119894+ 119891119868

ℎ

119894119891ℎ= 0 (120)

where 119888 = 119888(x) is a scalar function on119872

Theorem 27 Every twisted product Finsler manifold(119872

1times119891119872

2 119865) with isotropic mean Berwald curvature is a

weakly Berwald manifold

Proof Suppose that 119865 is isotropic mean Berwald twistedproduct Finsler metric Then differentiating (120) withrespect to V120574 gives us

119888 (119899 + 1) 1198912119865minus5V

120574119910119894= 0 (121)

Thus we conclude that 119888 = 0 This implies that 119865 reduces to aweakly Berwald metric

6 Locally Dually Flat Twisted ProductFinsler Manifolds

In [23] Amari and Nagaoka introduced the notion of duallyflat Riemannian metrics when they study the informationgeometry on Riemannian manifolds Information geometryhas emerged from investigating the geometrical structure ofa family of probability distributions and has been appliedsuccessfully to various areas including statistical inferencecontrol system theory andmultiterminal information theoryIn Finsler geometry Shen extends the notion of locally duallyflatness for Finsler metrics [17] Dually flat Finsler metricsform a special and valuable class of Finsler metrics in Finslerinformation geometry which play a very important role instudying flat Finsler information structure [24 25]

In this section we study locally dually flat twisted productFinsler metrics It is remarkable that a Finsler metric 119865 =119865(x y) on a manifold119872 is said to be locally dually flat if atany point there is a standard coordinate system (x119886 y119886) in119879119872such that it satisfies

12059721198652

120597x119887120597y119886y119887 = 2120597119865

2

120597x119886 (122)

In this case the coordinate (x119886) is called an adapted localcoordinate system Byusing (122) we can obtain the followinglemma

ISRN Geometry 11

Lemma 28 Let (1198721times119891119872

2 119865) be a twisted product Finsler

manifold Then 119865 is locally dually flat if and only if 1198651and 119865

2

satisfy in the following equations

12059721198652

1

120597119909119896120597119910119897119910119896= 2120597119865

2

1

120597119909119897+ 4119891119891

1198971198652

2 (123)

4119891119896V120573119910119896+ 119891

12059721198652

2

120597119906120572120597V120573V120572 + 4119891

120572V120573V120572 = 2119891

1205971198652

2

120597119906120573+ 4119891

1205731198652

2

(124)

Now let 119865 be a locally dually flat Finsler metric Takingderivative with respect to V120574 from (123) yields 119891

119897= 0 which

means that 119891 is a constant function on1198721 In this case the

relations (123) and (124) reduce to the following

12059721198652

1

120597119909119896120597119910119897119910119896= 2120597119865

2

1

120597119909119897 (125)

11989112059721198652

2

120597119906120572120597V120573V120572 + 4119891

120572V120573V120572 = 2119891

1205971198652

2

120597119906120573+ 4119891

1205731198652

2 (126)

By (125) we deduce that 1198651is locally dually flat

Now we assume that 1198651and 119865

2are locally dually flat

Finsler metrics Then we have

12059721198652

1

120597119909119896120597119910119897119910119896= 2120597119865

2

1

120597119909119897

12059721198652

2

120597119906120572120597V120573V120572 = 2

1205971198652

2

120597119906120573

(127)

By (127) we derive that (123) and (124) are hold if and only ifthe following hold

119891119897= 0 119891

120572V120573V120572 = 119891

1205731198652

2 (128)

Therefore we can conclude the following

Theorem 29 Let (1198721times119891119872

2 119865) be a twisted product Finsler

manifold

(i) If 119865 is locally dually flat then 1198651is locally dually flat 119891

is a function with respect to (119906120572) only and 1198652satisfies

in (126)(ii) If 119865

1and 119865

2are locally dually flat then 119865 is locally

dually flat if and only if 119891 is a function with respect(119906

120572) only and 119865

2satisfies in (128)

ByTheorem 29 we conclude the following

Corollary 30 There is not exist any locally dually flat propertwisted product Finsler manifold

References

[1] J K Beem P E Ehrlich and K L Easley Global LorentzianGeometry vol 202 of Monographs and Textbooks in Pure andApplied Mathematics Marcel Dekker New York NY USA 2ndedition 1996

[2] G Ganchev and V Mihova ldquoRiemannian manifolds of quasi-constant sectional curvaturesrdquo Journal fur die Reine und Ange-wandte Mathematik vol 522 pp 119ndash141 2000

[3] B OrsquoNeill Semi-Riemannian Geometry With Applications toRelativity vol 103 of Pure and Applied Mathematics AcademicPress New York NY USA 1983

[4] M Lohnherr and H Reckziegel ldquoOn ruled real hypersurfacesin complex space formsrdquo Geometriae Dedicata vol 74 no 3pp 267ndash286 1999

[5] B-Y Chen F Dillen L Verstraelen and L VranckenldquoLagrangian isometric immersions of a real-space-form119872119899

(119888)

into a complex-space-form119872119899(4119888)rdquoMathematical Proceedings

of the Cambridge Philosophical Society vol 124 no 1 pp 107ndash125 1998

[6] N Koike ldquoThe decomposition of curvature netted hypersur-facesrdquo Geometriae Dedicata vol 54 no 1 pp 1ndash11 1995

[7] B-y Chen Geometry of Submanifolds and its ApplicationsScience University of Tokyo Tokyo Japan 1981

[8] R Ponge and H Reckziegel ldquoTwisted products in pseudo-Riemannian geometryrdquo Geometriae Dedicata vol 48 no 1 pp15ndash25 1993

[9] B-Y Chen ldquoTwisted product CR-submanifolds in Kaehlermanifoldsrdquo Tamsui Oxford Journal of Mathematical Sciencesvol 16 no 2 pp 105ndash121 2000

[10] L Kozma I R Peter and H Shimada ldquoOn the twisted productof Finsler manifoldsrdquo Reports on Mathematical Physics vol 57no 3 pp 375ndash383 2006

[11] M Matsumoto and C Shibata ldquoOn semi-C-reducibility T-tensor= 0 and S4-likeness of Finsler spacesrdquo Journal of Math-ematics of Kyoto University vol 19 no 2 pp 301ndash314 1979

[12] B Najafi Z Shen and A Tayebi ldquoFinsler metrics of scalar flagcurvature with special non-Riemannian curvature propertiesrdquoGeometriae Dedicata vol 131 pp 87ndash97 2008

[13] E Peyghan andAHeydari ldquoConformal vector fields on tangentbundle of a Riemannian manifoldrdquo Journal of MathematicalAnalysis and Applications vol 347 no 1 pp 136ndash142 2008

[14] Z Shen Differential Geometry of Spray and Finsler SpacesKluwer Academic Publishers Dordrecht The Netherlands2001

[15] M Matsumoto and S-I Hojo ldquoA conclusive theorem on C-reducible Finsler spacesrdquo The Tensor Society vol 32 no 2 pp225ndash230 1978

[16] D Bao S-S Chern and Z Shen An Introduction to Riemann-Finsler Geometry vol 200 of Graduate Texts in MathematicsSpringer New York NY USA 2000

[17] Z Shen ldquoRiemann-Finsler geometry with applications to infor-mation geometryrdquo Chinese Annals of Mathematics B vol 27 no1 pp 73ndash94 2006

[18] Y Ichijyo ldquoFinsler manifolds modeled on a Minkowski spacerdquoJournal of Mathematics of Kyoto University vol 16 no 3 pp639ndash652 1976

[19] X Chen and Z Shen ldquoOn Douglas metricsrdquo PublicationesMathematicae Debrecen vol 66 no 3-4 pp 503ndash512 2005

[20] L Berwald ldquoUber Parallelubertragung in Raumen mit allge-meiner Massbestimmungrdquo Jahresbericht der DMVmdashDeutscheMathematiker-Vereinigung vol 34 pp 213ndash220 1926

[21] M Matsumoto ldquoOn Finsler spaces with Randersrsquo metric andspecial forms of important tensorsrdquo Journal of Mathematics ofKyoto University vol 14 pp 477ndash498 1974

12 ISRN Geometry

[22] A Tayebi and E Peyghan ldquoOn special Berwald metricsrdquoSymmetry Integrability and Geometry vol 6 article 008 9pages 2010

[23] S-I Amari and H NagaokaMethods of Information Geometryvol 191 of Translations of Mathematical Monographs AmericanMathematical Society Providence RI USA 2000

[24] X Cheng Z Shen and Y Zhou ldquoOn locally dually flat Finslermetricsrdquo International Journal ofMathematics vol 21 no 11 pp1531ndash1543 2010

[25] Q Xia ldquoOn a class of locally dually flat Finsler metrics ofisotropic flag curvaturerdquo Publicationes Mathematicae Debrecenvol 78 no 1 pp 169ndash190 2011

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

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Differential EquationsInternational Journal of

Volume 2014

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Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

6 ISRN Geometry

where

F119896119894119895= 119865

119896

119894119895minus (119872

119903

119895119862119896

119894119903+119872

119903

119894119862119896

119895119903minus119872

119903

ℎ119862119894119895119903119892119896ℎ) (57)

F119896119894120573= minusG119903

120573119862119896

119894119903= F119896

120573119894 (58)

F119896120572120573= minus119891119891

119896119892120572120573+ 119891

2119892119896ℎG120582

ℎ119862120572120573120582 (59)

F120574119894119895= 119891

minus2119892120574120582G119903

120582119862119894119895119903 (60)

F120574119894120573= 119891

minus1119891119894120575120574

120573minus G120572

119894119862120574

120572120573= F120574

120573119894 (61)

F120574120572120573= 119865

120574

120572120573+ 119873

120574

120572120573minus (119872

120583

120573119862120574

120572120583+119872

120583

120572119862120574

120573120583minus119872

120583

120582119862120572120573120583119892120574120582)

(62)

119865119896

119894119895=1

2119892119896ℎ(120575119892

ℎ119894

120575119909119895+

120575119892ℎ119895

120575119909119894minus

120575119892119894119895

120575119909ℎ)

119865120574

120572120573=1

2119892120574120582(120575119892

120582120572

120575119906120573+

120575119892120582120573

120575119906120572minus

120575119892120572120573

120575119906120582)

119872119903

119894= 119862

119903ℎ

119894119891119891

ℎ1198652

2

119872120583

120572= 119891

minus1(119862

120583120574

1205721198911205741198652

2+ 119891

119903119910119903120575120583

120572+ 119891

120574V120574120575120583

120572minus 119892

120583120574119891120574V120572+ 119891

120572V120583)

119873120574

120572120573= 119891

minus1(119891

120573120575120574

120572+ 119891

120572120575120574

120573minus 119891

120582119892120574120582119892120572120573)

(63)

Proof By using (55) we have

F119896119894119895=1

2119892119896ℎ(120575119905119892ℎ119894

120575119905119909119895+

120575119905119892ℎ119895

120575119905119909119894minus

120575119905119892119894119895

120575119905119909ℎ) (64)

Since 119892119894119895is a function with respect to (119909 119910) then by (25) and

(30) we obtain

120575119905119892ℎ119894

120575119905119909119895=120575119892

ℎ119894

120575119909119895minus 2119872

119903

119895119862ℎ119894119903 (65)

Interchanging 119894 119895 and ℎ in the previous equation gives us

120575119905119892ℎ119895

120575119905119909119894=

120575119892ℎ119895

120575119909119894minus 2119872

119903

119894119862ℎ119895119903

120575119905119892119894119895

120575119905119909ℎ=

120575119892119894119895

120575119909ℎminus 2119872

119903

ℎ119862119894119895119903

(66)

Putting these equation in (64) give us (57) In the similar waywe can prove the another relation

By using (i) of (23) and (57)ndash(62) we can conclude thefollowing

Lemma 9 Let (1198721times119891119872

2 119865) be a twisted product Finsler

manifold Then y119888F119886119887119888= G119886

119887 where F119886

119887119888and G119886

119887are defined by

(55) and (i) of (23) respectively

The Cartan torsion is one of the most important non-Riemannian quantity in Finsler geometry and it is first

introduced by Finsler and emphasized by Cartan whichmeasures a departure from a Riemannian manifold Moreprecisely a Finsler metric reduces to a Riemannian metricif and only if it has vanishing Cartan torsion The localcomponents of Cartan tensor field of the twisted Finslermanifold (119872

1times119891119872

2 119865) is defined by

C119886

119887119888=1

2g119886119890 120597g119887119890120597y119888 (67)

From this definition we conclude the following

Lemma 10 Let119862119896119894119895and119862120574

120572120573be the local components of Cartan

tensor field on1198721and119872

2 respectively Then one has

C119888

119886119887= (C119896

119894119895C119896

119894120573C119896

120572119895C119896

120572120573C120574

119894119895C120574

119894120573C120574

120572119895C120574

120572120573) (68)

where

C119896

119894119895=1

2119892119896ℎ120597119892

119894119895

120597119910ℎ= 119862

119896

119894119895

C120574

120572120573=1

2119892120574120582120597119892

120572120573

120597V120582= 119862

120574

120572120573

(69)

and C119896

119894120573= C119896

120572119895= C119896

120572120573= C120574

119894119895= C120574

119894120573= C120574

120572119895= 0

By using the Lemma 10 we can get the following

Corollary 11 Let (1198721times119891119872

2 119865) be a twisted product Finsler

manifoldThen (1198721times119891119872

2 119865) is a Riemannianmanifold if and

only if (1198721 119865

1) and (119872

2 119865

2) are Riemannian manifold

Various interesting special forms of Cartan tensors havebeen obtained by some Finslerians [11] The Finsler spaceshaving such special forms have been called C-reducible C2-like semi-C-reducible and so forth In [21] Matsumotointroduced the notion of C-reducible Finsler metrics andproved that any Randers metric is C-reducible Later onMatsumoto-Hojo proves that the converse is true too [15]

Here we define the Matsumoto twisted tensorM119886119887119888

for atwisted product Finsler manifold (119872

1times119891119872

2 119865) as follows

M119886119887119888= C

119886119887119888minus1

119899 + 1I

119886h119887119888+ I

119887h119886119888+ I

119888h119886119887 (70)

where I119886= g119887119888C

119886119887119888C

119886119887119888= g

119888119889C119889

119886119887 andh

119886119887= g

119886119887minus(1119865

2)y

119886y119887

By attention to the previous equation and relations

C119894119895119896= 119862

119894119895119896 C

120572120573120574= 119891

2119862120572120573120574 (71)

we obtain

M120572119895119896= minus

1

119899 + 1119868

120572(119892

119895119896minus1

1198652119910119895119910119896) minus1198912

1198652V120572(119868

119895119910119896+ 119868

119896119910119895)

(72)

Contracting the previous equation in 119910119895119910119896 gives us

119910119895119910119896M

120572119895119896= minus11989121198652

11198652

2

(119899 + 1) 1198652119868120572 (73)

ISRN Geometry 7

Similarly we obtain

V120582V120573M119894120573120582= minus11989121198652

11198652

2

(119899 + 1) 1198652119868119894 (74)

Therefore if M119894120573120582= M

120572119895119896= 0 then we get 119868

119894= 119868

120572= 0 that

is (1198721 119865

1) and (119872

2 119865

2) are Riemannian manifolds Thus we

have the following

Theorem 12 There is not exist any C-reducible twisted prod-uct Finsler manifold

Now we are going to consider semi-C-reducible twistedproduct Finsler manifold (119872

1times119891119872

2 119865) Let (119872

1times119891119872

2 119865) be

a semi-C-reducible twisted product Finsler manifold Thenwe have

C119886119887119888=119901

119899 + 1I

119886h119887119888+ I

119887h119886119888+ I

119888h119886119887 +119902

C2I119886I119887I119888 (75)

where C2= I119886I

119886and 119901 and 119902 are scalar function on119872

1times119891119872

2

with 119901 + 119902 = 1 This equation gives us

0 = C120572119895119896

=119901

119899 + 1119868

120572(119892

119895119896minus1

1198652119910119895119910119896) minus1198912

1198652V120572(119868

119895119910119896+ 119868

119896119910119895)

+119902

C2119868120572119868119895119868119896

(76)

Contractiing the previous equation with 119910119895119910119896 implies that

11990111989121198652

11198652

2119868120572= 0 (77)

Therefore we have 119901 = 0 or 119868120572= 0 If 119901 = 0 then

119865 is 1198622-like metric But if 119901 = 0 then 119868120572= 0 that is

1198652is Riemannian metric In this case with similar way

we conclude that 1198651is Riemannian metric But definition

119865 cannot be a Riemannian metric Therefore we have thefollowing

Theorem 13 Every semi-C-reducible twisted product Finslermanifold (119872

1times119891119872

2 119865) is a 1198622-like manifold

4 Riemannian Curvature

The Riemannian curvature of twisted product Finsler man-ifold (119872

1times119891119872

2 119865) with respect to Berwald connection is

given by

R 119886

119887 119888119889=120575119905F119886

119887119888

120575119905x119889minus120575119905F119886

119887119889

120575119905x119888+ F119886

119889119890F119890119887119888minus F119886

119888119890F119890119887119889 (78)

Lemma 14 Let (1198721times119891119872

2 119865) be a twisted product Finsler

manifold Then one has

R119886

119888119889= y119887R 119886

119887 119888119889 (79)

where R119886

119888119889and y119887R 119886

119887 119888119889are given by (50) and (78)

Proof By using (78) we have

y119887R 119894

119887 119896119897= y119887120575119905F119894

119887119896

120575119905x119897minus y119887120575119905F119894

119887119897

120575119905x119896+ y119887F119894

119897119890F119890119887119896 minus y119887F119894

119896119890F119890119887119897 (80)

By using Corollary 8 and Lemma 9 we obtain

y119887120575119905F119894

119887119896

120575119905x119897=120575119905G119894

119896

120575119905119909119897+ F119894

119895119896G119895

119897+ F119894

120573119896G120573

119897

y119887F119894119897119890F119890119887119896= F119894

119897ℎGℎ

119896+ F119894

119897120574G120574

119896

(81)

Interchanging 119894 and 119895 in the previous equation implies that

y119887120575119905F119894

119887119897

120575119905x119896=120575119905G119894

119897

120575119905119909119896+ F119894

119895119897G119895

119896+ F119894

120573119897G120573

119896

y119887F119894119896119890F119890119887119897= F119894

119896ℎGℎ

119897+ F119894

119896120574G120574

119897

(82)

Setting (81) and (82) in (80) gives us y119887R119894

119887 119896119897= R119894

119896119897 In the

similar way we can obtain this relation for another indices

Using (78) we can compute the Riemannian curvature ofa twisted product Finsler manifold

Lemma 15 Let (1198721times119891119872

2 119865) be a twisted product Finsler

manifoldThen the coefficients of Riemannian curvature are asfollows

R119894

119895 119896119897= 119877

119894

119895 119896119897

minus 119872119903

119897

120597119865119894

119895119896

120597119910119903+

120575119905119872

119894

119895119896

120575119905119909119897+119865

119894

119897ℎ119872

ℎ

119895119896+119872

119894

119897ℎ119865ℎ

119895119896minus119872

119894

119897ℎ119872

ℎ

119895119896

+ 119891minus2119892120572120574G119903

120572G119898

120574119862119894

119897119903119862119895119896119898 minusC

119896

119897

(83)

R 119894

120572 119896119897= minus

120575119905

120575119905119909119897(G119903

120572119862119894

119896119903) minus (119865

119894

119903119897minus119872

119894

119903119897)G119898

120572119862119903

119896119898

minus119891minus1G119903

120573119862119894

119897119903119891119896120575120573

120572+ G119903

120573G120583

119896119862119894

119897119903119862120573

120572120583 minus C

119896

119897

(84)

R 119894

119895 120573120582= minus

120575119905

120575119905119906120582(G119903

120573119862119894

119895119903) + G119898

120582G119897

120573119862119894

119903119898119862119903

119895119897

minus (119891119894119892120572120582minus 119891G120583

ℎ119892119894ℎ119862120572120582120583) (119891

119895120575120572

120573minus 119891G]

119895119862120572

120573])

minus C120573

120582

(85)

8 ISRN Geometry

R 119894

120572 120573119897=120575119905

120575119905119906120573(G119903

120572119862119894

119897119903) minus120575119905

120575119905119909119897119891 (119891

119894119892120572120573minus 119891G120582

ℎ119892119894ℎ119862120572120573120582)

minus G119898

120573G119904

120572119862119894

119903119898119862119903

119897119904+ (119891

119894119892120583120573minus 119891G120582

ℎ119892119894ℎ119862120583120573120582)

times (119891119897120575120583

120572minus 119891G]

119897119862120583

120572]) minus 119891119892119903ℎ(119865

119894

119903119897minus119872

119894

119903119897)

times (119891ℎ119892120572120573minus119891G120582

ℎ119862120572120573120582)minusG119903

120583119862119894

119897119903(119865

120583

120572120573+ 119873

120583

120572120573minus119872

120583

120572120573)

(86)

R 119894

119895 120573119897= minus

120575119905

120575119905119909119897(G119903

120573119862119894

119895119903) minus

120575119905

120575119905119906120573(119865

119894

119895119897minus119872

119894

119895119897)

minus (119865119894

119897119903minus119872

119894

119897119903)G119904

120573119862119903

119895119904minus 119891

minus1G119903

120572119862119894

119897119903

times (119891119895120575120572

120573minus 119891G120583

119895119862120572

120573120583) + G119904

120573119862119894

119903119904(119865

119903

119895119897minus119872

119903

119895119897)

+ 119891minus1G119903

120583119862119895119897119903(119891

119894120575120583

120573minus 119891G120582

ℎ119892119894ℎ119862120583

120573120582)

(87)

R 119894

120572 120573120582= minus

120575119905

120575119905119906120582(119891119891

119894119892120572120573minus 119891

2119892119894ℎG120583

ℎ119862120572120573120583) + 119891G119904

120582119862119894

119903119904

times(119891119903119892120572120573minus119891G120583

119897119862120572120573120583119892119903119897)minus119891 (119891

119894119892120582120583minus119891119892

119894ℎG120581

ℎ119862120582120583120581)

times (119865120583

120572120573+ 119873

120583

120572120573minus119872

120583

120572120573) minus C

120573

120582

(88)

R 120574

119895 119896119897=

120575119905

120575119905119909119897(119891

minus2119892120574120582G119903

120582119862119895119896119903) + 119891

minus2119892120574120582G119904

120582119862119897119903119904

times (119865119903

119895119896minus119872

119903

119895119896) + 119891

minus3G119903

120583119862119895119896119903(119891

119897119892120574120583minus 119891G120572

119897119862120574120583

120572)

minus C119896

119897

(89)

R 120574

119895 120573119897=120575119905

120575119905119909119897(119891

minus1119891119895120575120574

120573minus 119891G120572

119895119862120574

120572120573)

minus120575119905

120575119905119906120573(119891

minus2119892120574120582G119903

120582119862119895119897119903) minus 119891

minus2119892120574120582G119904

120582G119898

120573119862ℎ

119897119904119862ℎ119895119898

+ 119891minus2(119891

119897120575120574

120583minus 119891G120572

119897119862120574

120583120572) (119891

119895120575120583

120573minus 119891G]

119895119862120583

120573])

minus 119891minus1(119891

119903120575120574

120573minus 119891G120572

119903119862120574

120573120572) (119865

119903

119895119897minus119872

119903

119895119897)

minus 119891minus2119892120583120582G119903

120582119862119895119897119903(119865

120574

120573120583+ 119873

120574

120573120583minus119872

120574

120573120583)

(90)

R 120574

120572 120573119897=120575119905

120575119905119909119897(119865

120574

120572120573+ 119873

120574

120572120573minus119872

120574

120572120573)

minus120575119905

120575119905119906120573(119891

minus1119891119897120575120574

120572minus G120583

119897119862120574

120572120583)

minus 119891minus1119892120574120582G119904

120582119862ℎ

119897119904(119891

ℎ119892120572120573minus 119891G120583

ℎ119862120572120573120583)

+ 119891minus1(119891

119897120575120574

120583minus 119891G120581

119897119862120574

120583120581) (119865

120583

120572120573+ 119873

120583

120572120573minus119872

120583

120572120573)

+ 119891minus1G119904

120572119862119903

119897119904(119891

119903120575120574

120573minus 119891G120581

119903119862120574

120573120581)

minus 119891minus1(119891

119897120575120583

120572minus 119891G120581

119897119862120583

120572120581) (119865

120574

120573120583+ 119873

120574

120573120583minus119872

120574

120573120583)

(91)

R 120574

119895 120573120582=

120575119905

120575119905119906120582(119891

minus1119891119895120575120574

120573minus G120572

119895119862120574

120572120573)

+ 119891minus1(119865

120574

120572120582+ 119873

120574

120572120582minus119872

120574

120572120582) (119891

119895120575120572

120573minus 119891G]

119895119862120572

120573])

minus119891minus1G119898

120573119862119903

119895119898(119891

119903120575120574

120582minus 119891G120572

119903119862120574

120582120572) minus C

120573

120582

(92)

R 120574

120572 119896119897=

120575119905

120575119905119909119897(119891

minus1119891119896120575120574

120572minus G120583

119896119862120574

120572120583)

+ 119891minus2(119891

119897120575120574

120573minus 119891G120581

119897119862120574

120573120581) (119891

119896120575120573

120572minus 119891G]

119896119862120573

120572])

minus119891minus2G119904

120583G119898

120572119892120574120583119862ℎ

119897119904119862ℎ119896119898 minus C

119896

119897

(93)

R 120574

120572 120573120582= 119877

120574

120572 120573120582minus 119872

120581

120582

120597119865120574

120572120573

120597V120581+

120575119905119872

120574

120572120573

120575119905119906120582+119865

120574

120582120583119872

120583

120572120573+119872

120574

120582120583119865120583

120572120573

minus119872120574

120582120583119872

120583

120572120573+120575119905119873

120574

120572120582

120575119905119906120573+ 119865

120574

120573120583119873

120583

120572120582

+ 119873120574

120573120583119865120583

120572120582+ 119873

120574

120573120583119873

120583

120572120582+ 119873

120574

120582120583119872

120583

120572120573

+ 119873120583

120572120573119872

120574

120582120583+ (119892

119903119904120575120574

120582119891119904minus 119891119892

119903119904G120581

119904119862120574

120582120581)

times (119892120572120573119891119903minus 119891G120583

119903119862120572120573120583) minusC

120573

120582

(94)

where

119872119894

119895119896= 119872

119903

119896119862119894

119895119903+119872

119903

119895119862119894

119896119903minus119872

119903

ℎ119892119894ℎ119862119895119896119903

119872120574

120572120573= 119872

120583

120573119862120574

120572120583+119872

120583

120572119862120574

120573120583minus119872

120583

] 119892120574]119862120572120573120583

119873120574

120572120573= 119891

minus1(119891

120573120575120574

120572+ 119891

120572120575120574

120573minus 119891

120582119892120574120582119892120572120573)

(95)

and C119894

119895denotes the interchange of indices 119894 119895 and subtraction

ByTheorem 18 we have the following

Theorem 16 Let (1198721times119891119872

2 119865) be a flat twisted product

Finsler manifold and let (1198721 119865

1) be Riemannian If 119891 is a

function on1198722 only then (119872

1 119865

1) is locally flat

Similarly we get the following

Theorem 17 Let (1198721times119891119872

2 119865) be a flat twisted product

Finsler manifold and let (1198722 119865

2) be Riemannian If 119891 is a

function on 1198721 only then (119872

2 119865

2) is a space of positive

constant curvature ||119892119903119886119889119891||2

ISRN Geometry 9

Proof Since 1198722is Riemannain and 119891 is a function on 119872

1

then by (94) we obtain

R 120574

120572 120573120582= 119877

120574

120572 120573120582+1003817100381710038171003817119892119903119886119889119891

1003817100381710038171003817

2

(120575120574

120582119892120572120573minus 120575

120574

120573119892120572120582) (96)

Since (1198721times119891119872

2 119865) is flat then R120574

120572 120573120582= 0 Thus the proof is

complete

Theorem 18 Let (1198721times119891119872

2 119865) be a twisted product Rieman-

nian manifold and let 119891 be a function on 1198722 only Then

(1198721times119891119872

2 119865) is flat if and only if (119872

1 119865

1) is flat and the

Riemannian curvature of (1198722 119865

2) satisfies in the following

equation

119877120574

120572 120573120582= 120575119905119873

120574

120572120582

120575119905119906120573+ 119865

120574

120573120583119873

120583

120572120582+ 119873

120574

120573120583119865120583

120572120582+ 119873

120574

120573120583119873

120583

120572120582 minus C

120573

120582

(97)

5 Twisted Product Finsler Manifolds withNon-Riemannian Curvature Properties

There are several important non-Riemannian quantities suchas the Berwald curvature B the mean Berwald curvatureE and the Landsberg curvature L [22] They all vanishfor Riemannian metrics hence they are said to be non-Riemannian In this section we find some necessary and suf-ficient conditions underwhich a twisted product Riemannianmanifold are Berwaldian of isotropic Berwald curvatureof isotropic mean Berwald curvature First we prove thefollowing

Lemma 19 Let (1198721times119891119872

2 119865) be a twisted product Finsler

manifold Then the coefficients of Berwald curvature are asfollows

B120574

120572120573120582= 119861

120574

120572120573120582+ 119891

minus1(119862

120574]120582120572120573119891]119865

2

2+ 2119862

120574]120572120573119891]V120582

+ 2119862120574]120572120582119891]V120573 + 2119862

120574]120572119891]119892120582120573

+ 2119862120574]120582120573119891]V120572 + 2119862

120574]120573119891]119892120582120572

+ 2119862120574]120582119891]119892120572120573 minus 2119862120572120573120582119891

120574)

(98)

B119896

119894119895119897= 119861

119896

119894119895119897+ 119891119862

119896ℎ

119897119895119894119891ℎ1198652

2 (99)

B119896

119894120573119897= 2119891119862

119896ℎ

119894119897119891ℎV120573 (100)

B119896

120572120573119897= 2119891119892

120572120573119862119896ℎ

119897119891ℎ (101)

B119896

120572120573120582= minus 2119891119862

120572120573120582119891119896 (102)

B120574

119894120573120582= B120574

119894119895120582= B120574

119894119895119896= 0 (103)

Let (1198721times119891119872

2 119865) is a Berwald manifold Then we have

B119889

119886119887119888= 0 By using (102) we get

119862120572120573120582119891119896= 0 (104)

Multiplying this equation in 119892119896119903 we obtain

119862120572120573120582119891119903= 0 (105)

Thus if 119891 is not constant on1198721 then we have 119862

120572120573120582= 0 Also

from (101) we result that

119862119896ℎ

119897119891ℎ= 0 (106)

Differentiating this equation with respect to 119910119895 gives us

119862119896ℎ

119897119895119891ℎ= 0 (107)

Similarly we obtain

119862119896ℎ

119897119895119894119891ℎ= 0 (108)

Setting the last equation in (99) implies that 119861119896119894119895119897= 0 that

is (1198721 119865

1) is Berwaldian These explanations give us the

following theorem

Theorem 20 Let (1198721times119891119872

2 119865) be a twisted product Finsler

manifold and let119891 be not constant on1198721Then (119872

1times119891119872

2 119865)

is Berwaldian if and only if (1198721 119865

1) is Berwaldian (119872

2 119865

2) is

Riemannian and the equation 119862119896ℎ119897119891ℎ= 0 is hold

But if 119891 is constant on1198721 that is 119891

119894= 0 then we get the

following

Theorem 21 Let (1198721times119891119872

2 119865) be a twisted product Finsler

manifold and 119891 is constant on 1198721 Then (119872

1times119891119872

2 119865)

is Berwaldian if and only if (1198721 119865

1) is Berwaldian and

the Berwald curvature of (1198722 119865

2) satisfies in the following

equation

119861120574

120572120573120582= minus 119891

minus1(119862

120574]120573120572120582119891]119865

2

2+ 2119862

120574]120573120572119891]V120582 + 2119862

120574]120582120572119891]V120573

+ 2119862120574]120572119891]119892120582120573 + 2119862

120574]120573120582119891]V120572

+ 2119862120574]120573119891]119892120582120572 + 2119862

120574]120582119891]119892120572120573

minus 2119892120574]119862120572120573120582119891])

(109)

Here we consider twisted product Finsler manifold(119872

1times119891119872

2 119865) of isotropic Berwald curvature

Theorem 22 Every isotropic Berwald twisted product Finslermanifold (119872

1times119891119872

2 119865) is a Berwald manifold

Proof Let (1198721times119891119872

2 119865) be an isotropic Berwald manifold

Then we have

B119889

119886119887119888= 119888119865

minus1h119889

119886h119887119888+ h119889

119887h119886119888+ h119889

119888h119886119887+ 2C

119886119887119888y119889 (110)

where 119888 = 119888(x) is a function on119872 Setting 119886 = 119895 119887 = 119896 119888 = 119897and 119889 = 120574 and using (103) imply that

119888119865minus13

1198652119910119895119910119896119910119897V120574 minus V120574 (119910

119895119892119896119897+ 119910

119896119892119895119897+ 119910

119897119892119895119896) = 0 (111)

Multiplying the previous equation in 119910119895119910119896 we derive that119888119891

21198652

11198652

2= 0 Thus we have 119888 = 0 that is (119872

1times119891119872

2) is

Berwaldian

10 ISRN Geometry

Now we are going to study twisted product Finslermanifold of isotropicmean Berwald curvature For this workwe must compute the coefficients of mean Berwald curvatureof a twisted product Finsler manifold

Lemma 23 Let (1198721times119891119872

2 119865) be a twisted product Finsler

manifold Then the coefficients of mean Berwald curvature areas follows

E120572120573= 119864

120572120573+ 119891119892

120572120573119868ℎ119891ℎ+1

2119891119868

]120572120573119891]119865

2

2+ 119891

minus1119891]

times (119862120574]120572120573

V120574+ 119868

]120572V120573+ 119868

]120573V120572+ 119862

]120572120573+ 119868

]119892120572120573)

(112)

E119894119895= 119864

119894119895+1

2119891119868

ℎ

119895119894119891ℎ1198652

2 (113)

E119894120573= 119891119868

ℎ

119894119891ℎV120573 (114)

where 119864119894119895

and 119864120572120573

are the coefficients of mean Berwaldcurvature of (119872

1 119865

1) and (119872

2 119865

2) respectively

Proof By definition and Lemma 19 we get the proof

Theorem24 The twisted product Finslermanifold (1198721times119891119872

2

119865) is weakly Berwald if and only if (1198721 119865

1) is weakly Berwald

119868ℎ119891ℎ= 0 and the following hold

119864120572120573= minus

1

2119891119868

]120572120573119891]119865

2

2minus 119891

minus1119891]

times (119862120574]120572120573

V120574+ 119868

]120572V120573+ 119868

]120573V120572+ 119862

]120572120573+ 119868

]119892120572120573)

(115)

Proof If (1198721times119891119872

2) be a weakly Berwald manifold then we

have

E120572120573= E

119894119895= E

119894120573= 0 (116)

Thus by using (114) we result that 119868ℎ119894119891ℎ= 0 This equation

implies that

119868ℎ

119895119894119891ℎ= 0 119868

ℎ119891ℎ= 0 (117)

By setting these equations in (112) and (113) we conclude that119864119894119895= 0 and 119864

120572120573satisfies in (115)

Now if 119891 is constant on1198722 then (115) implies that 119864

120572120573=

0 Thus we conclude the following

Corollary 25 Let (1198721times119891119872

2 119865) be a twisted product Finsler

manifold and let 119891 be a function on 1198721 only Then

(1198721times119891119872

2 119865) is weakly Berwald if and only if (119872

1 119865

1) and

(1198722 119865

2) are weakly Berwald manifolds and 119868ℎ119891

ℎ= 0

Now we consider twisted product Finsler manifolds withisotropic mean Berwald curvature It is remarkable that as aconsequence of Lemma 23 we have the following

Lemma26 Twisted product Finslermanifold (1198721times119891119872

2 119865) is

isotropic mean Berwald manifold if and only if

119864120572120573+ 119891119892

120572120573119868ℎ119891ℎ+1

2119891119868

]120572120573119891]119865

2

2+ 119891

minus1119891]

times (119862120574]120572120573

V120574+ 119868

]120572V120573+ 119868

]120573V120572+ 119862

]120572120573+ 119868

]119892120572120573)

minus119899 + 1

2119888119891

2119865minus1(119892

120572120573minus1198912

1198652V120572V120573) = 0

(118)

119864119894119895+1

2119891119868

ℎ

119895119894119891ℎ1198652

2minus119899 + 1

2119888119865

minus1(119892

119894119895minus1

1198652119910119894119910119895) = 0 (119)

119888 (119899 + 1) 119865minus3119910119894+ 119891119868

ℎ

119894119891ℎ= 0 (120)

where 119888 = 119888(x) is a scalar function on119872

Theorem 27 Every twisted product Finsler manifold(119872

1times119891119872

2 119865) with isotropic mean Berwald curvature is a

weakly Berwald manifold

Proof Suppose that 119865 is isotropic mean Berwald twistedproduct Finsler metric Then differentiating (120) withrespect to V120574 gives us

119888 (119899 + 1) 1198912119865minus5V

120574119910119894= 0 (121)

Thus we conclude that 119888 = 0 This implies that 119865 reduces to aweakly Berwald metric

6 Locally Dually Flat Twisted ProductFinsler Manifolds

In [23] Amari and Nagaoka introduced the notion of duallyflat Riemannian metrics when they study the informationgeometry on Riemannian manifolds Information geometryhas emerged from investigating the geometrical structure ofa family of probability distributions and has been appliedsuccessfully to various areas including statistical inferencecontrol system theory andmultiterminal information theoryIn Finsler geometry Shen extends the notion of locally duallyflatness for Finsler metrics [17] Dually flat Finsler metricsform a special and valuable class of Finsler metrics in Finslerinformation geometry which play a very important role instudying flat Finsler information structure [24 25]

In this section we study locally dually flat twisted productFinsler metrics It is remarkable that a Finsler metric 119865 =119865(x y) on a manifold119872 is said to be locally dually flat if atany point there is a standard coordinate system (x119886 y119886) in119879119872such that it satisfies

12059721198652

120597x119887120597y119886y119887 = 2120597119865

2

120597x119886 (122)

In this case the coordinate (x119886) is called an adapted localcoordinate system Byusing (122) we can obtain the followinglemma

ISRN Geometry 11

Lemma 28 Let (1198721times119891119872

2 119865) be a twisted product Finsler

manifold Then 119865 is locally dually flat if and only if 1198651and 119865

2

satisfy in the following equations

12059721198652

1

120597119909119896120597119910119897119910119896= 2120597119865

2

1

120597119909119897+ 4119891119891

1198971198652

2 (123)

4119891119896V120573119910119896+ 119891

12059721198652

2

120597119906120572120597V120573V120572 + 4119891

120572V120573V120572 = 2119891

1205971198652

2

120597119906120573+ 4119891

1205731198652

2

(124)

Now let 119865 be a locally dually flat Finsler metric Takingderivative with respect to V120574 from (123) yields 119891

119897= 0 which

means that 119891 is a constant function on1198721 In this case the

relations (123) and (124) reduce to the following

12059721198652

1

120597119909119896120597119910119897119910119896= 2120597119865

2

1

120597119909119897 (125)

11989112059721198652

2

120597119906120572120597V120573V120572 + 4119891

120572V120573V120572 = 2119891

1205971198652

2

120597119906120573+ 4119891

1205731198652

2 (126)

By (125) we deduce that 1198651is locally dually flat

Now we assume that 1198651and 119865

2are locally dually flat

Finsler metrics Then we have

12059721198652

1

120597119909119896120597119910119897119910119896= 2120597119865

2

1

120597119909119897

12059721198652

2

120597119906120572120597V120573V120572 = 2

1205971198652

2

120597119906120573

(127)

By (127) we derive that (123) and (124) are hold if and only ifthe following hold

119891119897= 0 119891

120572V120573V120572 = 119891

1205731198652

2 (128)

Therefore we can conclude the following

Theorem 29 Let (1198721times119891119872

2 119865) be a twisted product Finsler

manifold

(i) If 119865 is locally dually flat then 1198651is locally dually flat 119891

is a function with respect to (119906120572) only and 1198652satisfies

in (126)(ii) If 119865

1and 119865

2are locally dually flat then 119865 is locally

dually flat if and only if 119891 is a function with respect(119906

120572) only and 119865

2satisfies in (128)

ByTheorem 29 we conclude the following

Corollary 30 There is not exist any locally dually flat propertwisted product Finsler manifold

References

[1] J K Beem P E Ehrlich and K L Easley Global LorentzianGeometry vol 202 of Monographs and Textbooks in Pure andApplied Mathematics Marcel Dekker New York NY USA 2ndedition 1996

[2] G Ganchev and V Mihova ldquoRiemannian manifolds of quasi-constant sectional curvaturesrdquo Journal fur die Reine und Ange-wandte Mathematik vol 522 pp 119ndash141 2000

[3] B OrsquoNeill Semi-Riemannian Geometry With Applications toRelativity vol 103 of Pure and Applied Mathematics AcademicPress New York NY USA 1983

[4] M Lohnherr and H Reckziegel ldquoOn ruled real hypersurfacesin complex space formsrdquo Geometriae Dedicata vol 74 no 3pp 267ndash286 1999

[5] B-Y Chen F Dillen L Verstraelen and L VranckenldquoLagrangian isometric immersions of a real-space-form119872119899

(119888)

into a complex-space-form119872119899(4119888)rdquoMathematical Proceedings

of the Cambridge Philosophical Society vol 124 no 1 pp 107ndash125 1998

[6] N Koike ldquoThe decomposition of curvature netted hypersur-facesrdquo Geometriae Dedicata vol 54 no 1 pp 1ndash11 1995

[7] B-y Chen Geometry of Submanifolds and its ApplicationsScience University of Tokyo Tokyo Japan 1981

[8] R Ponge and H Reckziegel ldquoTwisted products in pseudo-Riemannian geometryrdquo Geometriae Dedicata vol 48 no 1 pp15ndash25 1993

[9] B-Y Chen ldquoTwisted product CR-submanifolds in Kaehlermanifoldsrdquo Tamsui Oxford Journal of Mathematical Sciencesvol 16 no 2 pp 105ndash121 2000

[10] L Kozma I R Peter and H Shimada ldquoOn the twisted productof Finsler manifoldsrdquo Reports on Mathematical Physics vol 57no 3 pp 375ndash383 2006

[11] M Matsumoto and C Shibata ldquoOn semi-C-reducibility T-tensor= 0 and S4-likeness of Finsler spacesrdquo Journal of Math-ematics of Kyoto University vol 19 no 2 pp 301ndash314 1979

[12] B Najafi Z Shen and A Tayebi ldquoFinsler metrics of scalar flagcurvature with special non-Riemannian curvature propertiesrdquoGeometriae Dedicata vol 131 pp 87ndash97 2008

[13] E Peyghan andAHeydari ldquoConformal vector fields on tangentbundle of a Riemannian manifoldrdquo Journal of MathematicalAnalysis and Applications vol 347 no 1 pp 136ndash142 2008

[14] Z Shen Differential Geometry of Spray and Finsler SpacesKluwer Academic Publishers Dordrecht The Netherlands2001

[15] M Matsumoto and S-I Hojo ldquoA conclusive theorem on C-reducible Finsler spacesrdquo The Tensor Society vol 32 no 2 pp225ndash230 1978

[16] D Bao S-S Chern and Z Shen An Introduction to Riemann-Finsler Geometry vol 200 of Graduate Texts in MathematicsSpringer New York NY USA 2000

[17] Z Shen ldquoRiemann-Finsler geometry with applications to infor-mation geometryrdquo Chinese Annals of Mathematics B vol 27 no1 pp 73ndash94 2006

[18] Y Ichijyo ldquoFinsler manifolds modeled on a Minkowski spacerdquoJournal of Mathematics of Kyoto University vol 16 no 3 pp639ndash652 1976

[19] X Chen and Z Shen ldquoOn Douglas metricsrdquo PublicationesMathematicae Debrecen vol 66 no 3-4 pp 503ndash512 2005

[20] L Berwald ldquoUber Parallelubertragung in Raumen mit allge-meiner Massbestimmungrdquo Jahresbericht der DMVmdashDeutscheMathematiker-Vereinigung vol 34 pp 213ndash220 1926

[21] M Matsumoto ldquoOn Finsler spaces with Randersrsquo metric andspecial forms of important tensorsrdquo Journal of Mathematics ofKyoto University vol 14 pp 477ndash498 1974

12 ISRN Geometry

[22] A Tayebi and E Peyghan ldquoOn special Berwald metricsrdquoSymmetry Integrability and Geometry vol 6 article 008 9pages 2010

[23] S-I Amari and H NagaokaMethods of Information Geometryvol 191 of Translations of Mathematical Monographs AmericanMathematical Society Providence RI USA 2000

[24] X Cheng Z Shen and Y Zhou ldquoOn locally dually flat Finslermetricsrdquo International Journal ofMathematics vol 21 no 11 pp1531ndash1543 2010

[25] Q Xia ldquoOn a class of locally dually flat Finsler metrics ofisotropic flag curvaturerdquo Publicationes Mathematicae Debrecenvol 78 no 1 pp 169ndash190 2011

Submit your manuscripts athttpwwwhindawicom

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Mathematical Problems in Engineering

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Differential EquationsInternational Journal of

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Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Decision SciencesAdvances in

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Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

ISRN Geometry 7

Similarly we obtain

V120582V120573M119894120573120582= minus11989121198652

11198652

2

(119899 + 1) 1198652119868119894 (74)

Therefore if M119894120573120582= M

120572119895119896= 0 then we get 119868

119894= 119868

120572= 0 that

is (1198721 119865

1) and (119872

2 119865

2) are Riemannian manifolds Thus we

have the following

Theorem 12 There is not exist any C-reducible twisted prod-uct Finsler manifold

Now we are going to consider semi-C-reducible twistedproduct Finsler manifold (119872

1times119891119872

2 119865) Let (119872

1times119891119872

2 119865) be

a semi-C-reducible twisted product Finsler manifold Thenwe have

C119886119887119888=119901

119899 + 1I

119886h119887119888+ I

119887h119886119888+ I

119888h119886119887 +119902

C2I119886I119887I119888 (75)

where C2= I119886I

119886and 119901 and 119902 are scalar function on119872

1times119891119872

2

with 119901 + 119902 = 1 This equation gives us

0 = C120572119895119896

=119901

119899 + 1119868

120572(119892

119895119896minus1

1198652119910119895119910119896) minus1198912

1198652V120572(119868

119895119910119896+ 119868

119896119910119895)

+119902

C2119868120572119868119895119868119896

(76)

Contractiing the previous equation with 119910119895119910119896 implies that

11990111989121198652

11198652

2119868120572= 0 (77)

Therefore we have 119901 = 0 or 119868120572= 0 If 119901 = 0 then

119865 is 1198622-like metric But if 119901 = 0 then 119868120572= 0 that is

1198652is Riemannian metric In this case with similar way

we conclude that 1198651is Riemannian metric But definition

119865 cannot be a Riemannian metric Therefore we have thefollowing

Theorem 13 Every semi-C-reducible twisted product Finslermanifold (119872

1times119891119872

2 119865) is a 1198622-like manifold

4 Riemannian Curvature

The Riemannian curvature of twisted product Finsler man-ifold (119872

1times119891119872

2 119865) with respect to Berwald connection is

given by

R 119886

119887 119888119889=120575119905F119886

119887119888

120575119905x119889minus120575119905F119886

119887119889

120575119905x119888+ F119886

119889119890F119890119887119888minus F119886

119888119890F119890119887119889 (78)

Lemma 14 Let (1198721times119891119872

2 119865) be a twisted product Finsler

manifold Then one has

R119886

119888119889= y119887R 119886

119887 119888119889 (79)

where R119886

119888119889and y119887R 119886

119887 119888119889are given by (50) and (78)

Proof By using (78) we have

y119887R 119894

119887 119896119897= y119887120575119905F119894

119887119896

120575119905x119897minus y119887120575119905F119894

119887119897

120575119905x119896+ y119887F119894

119897119890F119890119887119896 minus y119887F119894

119896119890F119890119887119897 (80)

By using Corollary 8 and Lemma 9 we obtain

y119887120575119905F119894

119887119896

120575119905x119897=120575119905G119894

119896

120575119905119909119897+ F119894

119895119896G119895

119897+ F119894

120573119896G120573

119897

y119887F119894119897119890F119890119887119896= F119894

119897ℎGℎ

119896+ F119894

119897120574G120574

119896

(81)

Interchanging 119894 and 119895 in the previous equation implies that

y119887120575119905F119894

119887119897

120575119905x119896=120575119905G119894

119897

120575119905119909119896+ F119894

119895119897G119895

119896+ F119894

120573119897G120573

119896

y119887F119894119896119890F119890119887119897= F119894

119896ℎGℎ

119897+ F119894

119896120574G120574

119897

(82)

Setting (81) and (82) in (80) gives us y119887R119894

119887 119896119897= R119894

119896119897 In the

similar way we can obtain this relation for another indices

Using (78) we can compute the Riemannian curvature ofa twisted product Finsler manifold

Lemma 15 Let (1198721times119891119872

2 119865) be a twisted product Finsler

manifoldThen the coefficients of Riemannian curvature are asfollows

R119894

119895 119896119897= 119877

119894

119895 119896119897

minus 119872119903

119897

120597119865119894

119895119896

120597119910119903+

120575119905119872

119894

119895119896

120575119905119909119897+119865

119894

119897ℎ119872

ℎ

119895119896+119872

119894

119897ℎ119865ℎ

119895119896minus119872

119894

119897ℎ119872

ℎ

119895119896

+ 119891minus2119892120572120574G119903

120572G119898

120574119862119894

119897119903119862119895119896119898 minusC

119896

119897

(83)

R 119894

120572 119896119897= minus

120575119905

120575119905119909119897(G119903

120572119862119894

119896119903) minus (119865

119894

119903119897minus119872

119894

119903119897)G119898

120572119862119903

119896119898

minus119891minus1G119903

120573119862119894

119897119903119891119896120575120573

120572+ G119903

120573G120583

119896119862119894

119897119903119862120573

120572120583 minus C

119896

119897

(84)

R 119894

119895 120573120582= minus

120575119905

120575119905119906120582(G119903

120573119862119894

119895119903) + G119898

120582G119897

120573119862119894

119903119898119862119903

119895119897

minus (119891119894119892120572120582minus 119891G120583

ℎ119892119894ℎ119862120572120582120583) (119891

119895120575120572

120573minus 119891G]

119895119862120572

120573])

minus C120573

120582

(85)

8 ISRN Geometry

R 119894

120572 120573119897=120575119905

120575119905119906120573(G119903

120572119862119894

119897119903) minus120575119905

120575119905119909119897119891 (119891

119894119892120572120573minus 119891G120582

ℎ119892119894ℎ119862120572120573120582)

minus G119898

120573G119904

120572119862119894

119903119898119862119903

119897119904+ (119891

119894119892120583120573minus 119891G120582

ℎ119892119894ℎ119862120583120573120582)

times (119891119897120575120583

120572minus 119891G]

119897119862120583

120572]) minus 119891119892119903ℎ(119865

119894

119903119897minus119872

119894

119903119897)

times (119891ℎ119892120572120573minus119891G120582

ℎ119862120572120573120582)minusG119903

120583119862119894

119897119903(119865

120583

120572120573+ 119873

120583

120572120573minus119872

120583

120572120573)

(86)

R 119894

119895 120573119897= minus

120575119905

120575119905119909119897(G119903

120573119862119894

119895119903) minus

120575119905

120575119905119906120573(119865

119894

119895119897minus119872

119894

119895119897)

minus (119865119894

119897119903minus119872

119894

119897119903)G119904

120573119862119903

119895119904minus 119891

minus1G119903

120572119862119894

119897119903

times (119891119895120575120572

120573minus 119891G120583

119895119862120572

120573120583) + G119904

120573119862119894

119903119904(119865

119903

119895119897minus119872

119903

119895119897)

+ 119891minus1G119903

120583119862119895119897119903(119891

119894120575120583

120573minus 119891G120582

ℎ119892119894ℎ119862120583

120573120582)

(87)

R 119894

120572 120573120582= minus

120575119905

120575119905119906120582(119891119891

119894119892120572120573minus 119891

2119892119894ℎG120583

ℎ119862120572120573120583) + 119891G119904

120582119862119894

119903119904

times(119891119903119892120572120573minus119891G120583

119897119862120572120573120583119892119903119897)minus119891 (119891

119894119892120582120583minus119891119892

119894ℎG120581

ℎ119862120582120583120581)

times (119865120583

120572120573+ 119873

120583

120572120573minus119872

120583

120572120573) minus C

120573

120582

(88)

R 120574

119895 119896119897=

120575119905

120575119905119909119897(119891

minus2119892120574120582G119903

120582119862119895119896119903) + 119891

minus2119892120574120582G119904

120582119862119897119903119904

times (119865119903

119895119896minus119872

119903

119895119896) + 119891

minus3G119903

120583119862119895119896119903(119891

119897119892120574120583minus 119891G120572

119897119862120574120583

120572)

minus C119896

119897

(89)

R 120574

119895 120573119897=120575119905

120575119905119909119897(119891

minus1119891119895120575120574

120573minus 119891G120572

119895119862120574

120572120573)

minus120575119905

120575119905119906120573(119891

minus2119892120574120582G119903

120582119862119895119897119903) minus 119891

minus2119892120574120582G119904

120582G119898

120573119862ℎ

119897119904119862ℎ119895119898

+ 119891minus2(119891

119897120575120574

120583minus 119891G120572

119897119862120574

120583120572) (119891

119895120575120583

120573minus 119891G]

119895119862120583

120573])

minus 119891minus1(119891

119903120575120574

120573minus 119891G120572

119903119862120574

120573120572) (119865

119903

119895119897minus119872

119903

119895119897)

minus 119891minus2119892120583120582G119903

120582119862119895119897119903(119865

120574

120573120583+ 119873

120574

120573120583minus119872

120574

120573120583)

(90)

R 120574

120572 120573119897=120575119905

120575119905119909119897(119865

120574

120572120573+ 119873

120574

120572120573minus119872

120574

120572120573)

minus120575119905

120575119905119906120573(119891

minus1119891119897120575120574

120572minus G120583

119897119862120574

120572120583)

minus 119891minus1119892120574120582G119904

120582119862ℎ

119897119904(119891

ℎ119892120572120573minus 119891G120583

ℎ119862120572120573120583)

+ 119891minus1(119891

119897120575120574

120583minus 119891G120581

119897119862120574

120583120581) (119865

120583

120572120573+ 119873

120583

120572120573minus119872

120583

120572120573)

+ 119891minus1G119904

120572119862119903

119897119904(119891

119903120575120574

120573minus 119891G120581

119903119862120574

120573120581)

minus 119891minus1(119891

119897120575120583

120572minus 119891G120581

119897119862120583

120572120581) (119865

120574

120573120583+ 119873

120574

120573120583minus119872

120574

120573120583)

(91)

R 120574

119895 120573120582=

120575119905

120575119905119906120582(119891

minus1119891119895120575120574

120573minus G120572

119895119862120574

120572120573)

+ 119891minus1(119865

120574

120572120582+ 119873

120574

120572120582minus119872

120574

120572120582) (119891

119895120575120572

120573minus 119891G]

119895119862120572

120573])

minus119891minus1G119898

120573119862119903

119895119898(119891

119903120575120574

120582minus 119891G120572

119903119862120574

120582120572) minus C

120573

120582

(92)

R 120574

120572 119896119897=

120575119905

120575119905119909119897(119891

minus1119891119896120575120574

120572minus G120583

119896119862120574

120572120583)

+ 119891minus2(119891

119897120575120574

120573minus 119891G120581

119897119862120574

120573120581) (119891

119896120575120573

120572minus 119891G]

119896119862120573

120572])

minus119891minus2G119904

120583G119898

120572119892120574120583119862ℎ

119897119904119862ℎ119896119898 minus C

119896

119897

(93)

R 120574

120572 120573120582= 119877

120574

120572 120573120582minus 119872

120581

120582

120597119865120574

120572120573

120597V120581+

120575119905119872

120574

120572120573

120575119905119906120582+119865

120574

120582120583119872

120583

120572120573+119872

120574

120582120583119865120583

120572120573

minus119872120574

120582120583119872

120583

120572120573+120575119905119873

120574

120572120582

120575119905119906120573+ 119865

120574

120573120583119873

120583

120572120582

+ 119873120574

120573120583119865120583

120572120582+ 119873

120574

120573120583119873

120583

120572120582+ 119873

120574

120582120583119872

120583

120572120573

+ 119873120583

120572120573119872

120574

120582120583+ (119892

119903119904120575120574

120582119891119904minus 119891119892

119903119904G120581

119904119862120574

120582120581)

times (119892120572120573119891119903minus 119891G120583

119903119862120572120573120583) minusC

120573

120582

(94)

where

119872119894

119895119896= 119872

119903

119896119862119894

119895119903+119872

119903

119895119862119894

119896119903minus119872

119903

ℎ119892119894ℎ119862119895119896119903

119872120574

120572120573= 119872

120583

120573119862120574

120572120583+119872

120583

120572119862120574

120573120583minus119872

120583

] 119892120574]119862120572120573120583

119873120574

120572120573= 119891

minus1(119891

120573120575120574

120572+ 119891

120572120575120574

120573minus 119891

120582119892120574120582119892120572120573)

(95)

and C119894

119895denotes the interchange of indices 119894 119895 and subtraction

ByTheorem 18 we have the following

Theorem 16 Let (1198721times119891119872

2 119865) be a flat twisted product

Finsler manifold and let (1198721 119865

1) be Riemannian If 119891 is a

function on1198722 only then (119872

1 119865

1) is locally flat

Similarly we get the following

Theorem 17 Let (1198721times119891119872

2 119865) be a flat twisted product

Finsler manifold and let (1198722 119865

2) be Riemannian If 119891 is a

function on 1198721 only then (119872

2 119865

2) is a space of positive

constant curvature ||119892119903119886119889119891||2

ISRN Geometry 9

Proof Since 1198722is Riemannain and 119891 is a function on 119872

1

then by (94) we obtain

R 120574

120572 120573120582= 119877

120574

120572 120573120582+1003817100381710038171003817119892119903119886119889119891

1003817100381710038171003817

2

(120575120574

120582119892120572120573minus 120575

120574

120573119892120572120582) (96)

Since (1198721times119891119872

2 119865) is flat then R120574

120572 120573120582= 0 Thus the proof is

complete

Theorem 18 Let (1198721times119891119872

2 119865) be a twisted product Rieman-

nian manifold and let 119891 be a function on 1198722 only Then

(1198721times119891119872

2 119865) is flat if and only if (119872

1 119865

1) is flat and the

Riemannian curvature of (1198722 119865

2) satisfies in the following

equation

119877120574

120572 120573120582= 120575119905119873

120574

120572120582

120575119905119906120573+ 119865

120574

120573120583119873

120583

120572120582+ 119873

120574

120573120583119865120583

120572120582+ 119873

120574

120573120583119873

120583

120572120582 minus C

120573

120582

(97)

5 Twisted Product Finsler Manifolds withNon-Riemannian Curvature Properties

There are several important non-Riemannian quantities suchas the Berwald curvature B the mean Berwald curvatureE and the Landsberg curvature L [22] They all vanishfor Riemannian metrics hence they are said to be non-Riemannian In this section we find some necessary and suf-ficient conditions underwhich a twisted product Riemannianmanifold are Berwaldian of isotropic Berwald curvatureof isotropic mean Berwald curvature First we prove thefollowing

Lemma 19 Let (1198721times119891119872

2 119865) be a twisted product Finsler

manifold Then the coefficients of Berwald curvature are asfollows

B120574

120572120573120582= 119861

120574

120572120573120582+ 119891

minus1(119862

120574]120582120572120573119891]119865

2

2+ 2119862

120574]120572120573119891]V120582

+ 2119862120574]120572120582119891]V120573 + 2119862

120574]120572119891]119892120582120573

+ 2119862120574]120582120573119891]V120572 + 2119862

120574]120573119891]119892120582120572

+ 2119862120574]120582119891]119892120572120573 minus 2119862120572120573120582119891

120574)

(98)

B119896

119894119895119897= 119861

119896

119894119895119897+ 119891119862

119896ℎ

119897119895119894119891ℎ1198652

2 (99)

B119896

119894120573119897= 2119891119862

119896ℎ

119894119897119891ℎV120573 (100)

B119896

120572120573119897= 2119891119892

120572120573119862119896ℎ

119897119891ℎ (101)

B119896

120572120573120582= minus 2119891119862

120572120573120582119891119896 (102)

B120574

119894120573120582= B120574

119894119895120582= B120574

119894119895119896= 0 (103)

Let (1198721times119891119872

2 119865) is a Berwald manifold Then we have

B119889

119886119887119888= 0 By using (102) we get

119862120572120573120582119891119896= 0 (104)

Multiplying this equation in 119892119896119903 we obtain

119862120572120573120582119891119903= 0 (105)

Thus if 119891 is not constant on1198721 then we have 119862

120572120573120582= 0 Also

from (101) we result that

119862119896ℎ

119897119891ℎ= 0 (106)

Differentiating this equation with respect to 119910119895 gives us

119862119896ℎ

119897119895119891ℎ= 0 (107)

Similarly we obtain

119862119896ℎ

119897119895119894119891ℎ= 0 (108)

Setting the last equation in (99) implies that 119861119896119894119895119897= 0 that

is (1198721 119865

1) is Berwaldian These explanations give us the

following theorem

Theorem 20 Let (1198721times119891119872

2 119865) be a twisted product Finsler

manifold and let119891 be not constant on1198721Then (119872

1times119891119872

2 119865)

is Berwaldian if and only if (1198721 119865

1) is Berwaldian (119872

2 119865

2) is

Riemannian and the equation 119862119896ℎ119897119891ℎ= 0 is hold

But if 119891 is constant on1198721 that is 119891

119894= 0 then we get the

following

Theorem 21 Let (1198721times119891119872

2 119865) be a twisted product Finsler

manifold and 119891 is constant on 1198721 Then (119872

1times119891119872

2 119865)

is Berwaldian if and only if (1198721 119865

1) is Berwaldian and

the Berwald curvature of (1198722 119865

2) satisfies in the following

equation

119861120574

120572120573120582= minus 119891

minus1(119862

120574]120573120572120582119891]119865

2

2+ 2119862

120574]120573120572119891]V120582 + 2119862

120574]120582120572119891]V120573

+ 2119862120574]120572119891]119892120582120573 + 2119862

120574]120573120582119891]V120572

+ 2119862120574]120573119891]119892120582120572 + 2119862

120574]120582119891]119892120572120573

minus 2119892120574]119862120572120573120582119891])

(109)

Here we consider twisted product Finsler manifold(119872

1times119891119872

2 119865) of isotropic Berwald curvature

Theorem 22 Every isotropic Berwald twisted product Finslermanifold (119872

1times119891119872

2 119865) is a Berwald manifold

Proof Let (1198721times119891119872

2 119865) be an isotropic Berwald manifold

Then we have

B119889

119886119887119888= 119888119865

minus1h119889

119886h119887119888+ h119889

119887h119886119888+ h119889

119888h119886119887+ 2C

119886119887119888y119889 (110)

where 119888 = 119888(x) is a function on119872 Setting 119886 = 119895 119887 = 119896 119888 = 119897and 119889 = 120574 and using (103) imply that

119888119865minus13

1198652119910119895119910119896119910119897V120574 minus V120574 (119910

119895119892119896119897+ 119910

119896119892119895119897+ 119910

119897119892119895119896) = 0 (111)

Multiplying the previous equation in 119910119895119910119896 we derive that119888119891

21198652

11198652

2= 0 Thus we have 119888 = 0 that is (119872

1times119891119872

2) is

Berwaldian

10 ISRN Geometry

Now we are going to study twisted product Finslermanifold of isotropicmean Berwald curvature For this workwe must compute the coefficients of mean Berwald curvatureof a twisted product Finsler manifold

Lemma 23 Let (1198721times119891119872

2 119865) be a twisted product Finsler

manifold Then the coefficients of mean Berwald curvature areas follows

E120572120573= 119864

120572120573+ 119891119892

120572120573119868ℎ119891ℎ+1

2119891119868

]120572120573119891]119865

2

2+ 119891

minus1119891]

times (119862120574]120572120573

V120574+ 119868

]120572V120573+ 119868

]120573V120572+ 119862

]120572120573+ 119868

]119892120572120573)

(112)

E119894119895= 119864

119894119895+1

2119891119868

ℎ

119895119894119891ℎ1198652

2 (113)

E119894120573= 119891119868

ℎ

119894119891ℎV120573 (114)

where 119864119894119895

and 119864120572120573

are the coefficients of mean Berwaldcurvature of (119872

1 119865

1) and (119872

2 119865

2) respectively

Proof By definition and Lemma 19 we get the proof

Theorem24 The twisted product Finslermanifold (1198721times119891119872

2

119865) is weakly Berwald if and only if (1198721 119865

1) is weakly Berwald

119868ℎ119891ℎ= 0 and the following hold

119864120572120573= minus

1

2119891119868

]120572120573119891]119865

2

2minus 119891

minus1119891]

times (119862120574]120572120573

V120574+ 119868

]120572V120573+ 119868

]120573V120572+ 119862

]120572120573+ 119868

]119892120572120573)

(115)

Proof If (1198721times119891119872

2) be a weakly Berwald manifold then we

have

E120572120573= E

119894119895= E

119894120573= 0 (116)

Thus by using (114) we result that 119868ℎ119894119891ℎ= 0 This equation

implies that

119868ℎ

119895119894119891ℎ= 0 119868

ℎ119891ℎ= 0 (117)

By setting these equations in (112) and (113) we conclude that119864119894119895= 0 and 119864

120572120573satisfies in (115)

Now if 119891 is constant on1198722 then (115) implies that 119864

120572120573=

0 Thus we conclude the following

Corollary 25 Let (1198721times119891119872

2 119865) be a twisted product Finsler

manifold and let 119891 be a function on 1198721 only Then

(1198721times119891119872

2 119865) is weakly Berwald if and only if (119872

1 119865

1) and

(1198722 119865

2) are weakly Berwald manifolds and 119868ℎ119891

ℎ= 0

Now we consider twisted product Finsler manifolds withisotropic mean Berwald curvature It is remarkable that as aconsequence of Lemma 23 we have the following

Lemma26 Twisted product Finslermanifold (1198721times119891119872

2 119865) is

isotropic mean Berwald manifold if and only if

119864120572120573+ 119891119892

120572120573119868ℎ119891ℎ+1

2119891119868

]120572120573119891]119865

2

2+ 119891

minus1119891]

times (119862120574]120572120573

V120574+ 119868

]120572V120573+ 119868

]120573V120572+ 119862

]120572120573+ 119868

]119892120572120573)

minus119899 + 1

2119888119891

2119865minus1(119892

120572120573minus1198912

1198652V120572V120573) = 0

(118)

119864119894119895+1

2119891119868

ℎ

119895119894119891ℎ1198652

2minus119899 + 1

2119888119865

minus1(119892

119894119895minus1

1198652119910119894119910119895) = 0 (119)

119888 (119899 + 1) 119865minus3119910119894+ 119891119868

ℎ

119894119891ℎ= 0 (120)

where 119888 = 119888(x) is a scalar function on119872

Theorem 27 Every twisted product Finsler manifold(119872

1times119891119872

2 119865) with isotropic mean Berwald curvature is a

weakly Berwald manifold

Proof Suppose that 119865 is isotropic mean Berwald twistedproduct Finsler metric Then differentiating (120) withrespect to V120574 gives us

119888 (119899 + 1) 1198912119865minus5V

120574119910119894= 0 (121)

Thus we conclude that 119888 = 0 This implies that 119865 reduces to aweakly Berwald metric

6 Locally Dually Flat Twisted ProductFinsler Manifolds

In [23] Amari and Nagaoka introduced the notion of duallyflat Riemannian metrics when they study the informationgeometry on Riemannian manifolds Information geometryhas emerged from investigating the geometrical structure ofa family of probability distributions and has been appliedsuccessfully to various areas including statistical inferencecontrol system theory andmultiterminal information theoryIn Finsler geometry Shen extends the notion of locally duallyflatness for Finsler metrics [17] Dually flat Finsler metricsform a special and valuable class of Finsler metrics in Finslerinformation geometry which play a very important role instudying flat Finsler information structure [24 25]

In this section we study locally dually flat twisted productFinsler metrics It is remarkable that a Finsler metric 119865 =119865(x y) on a manifold119872 is said to be locally dually flat if atany point there is a standard coordinate system (x119886 y119886) in119879119872such that it satisfies

12059721198652

120597x119887120597y119886y119887 = 2120597119865

2

120597x119886 (122)

In this case the coordinate (x119886) is called an adapted localcoordinate system Byusing (122) we can obtain the followinglemma

ISRN Geometry 11

Lemma 28 Let (1198721times119891119872

2 119865) be a twisted product Finsler

manifold Then 119865 is locally dually flat if and only if 1198651and 119865

2

satisfy in the following equations

12059721198652

1

120597119909119896120597119910119897119910119896= 2120597119865

2

1

120597119909119897+ 4119891119891

1198971198652

2 (123)

4119891119896V120573119910119896+ 119891

12059721198652

2

120597119906120572120597V120573V120572 + 4119891

120572V120573V120572 = 2119891

1205971198652

2

120597119906120573+ 4119891

1205731198652

2

(124)

Now let 119865 be a locally dually flat Finsler metric Takingderivative with respect to V120574 from (123) yields 119891

119897= 0 which

means that 119891 is a constant function on1198721 In this case the

relations (123) and (124) reduce to the following

12059721198652

1

120597119909119896120597119910119897119910119896= 2120597119865

2

1

120597119909119897 (125)

11989112059721198652

2

120597119906120572120597V120573V120572 + 4119891

120572V120573V120572 = 2119891

1205971198652

2

120597119906120573+ 4119891

1205731198652

2 (126)

By (125) we deduce that 1198651is locally dually flat

Now we assume that 1198651and 119865

2are locally dually flat

Finsler metrics Then we have

12059721198652

1

120597119909119896120597119910119897119910119896= 2120597119865

2

1

120597119909119897

12059721198652

2

120597119906120572120597V120573V120572 = 2

1205971198652

2

120597119906120573

(127)

By (127) we derive that (123) and (124) are hold if and only ifthe following hold

119891119897= 0 119891

120572V120573V120572 = 119891

1205731198652

2 (128)

Therefore we can conclude the following

Theorem 29 Let (1198721times119891119872

2 119865) be a twisted product Finsler

manifold

(i) If 119865 is locally dually flat then 1198651is locally dually flat 119891

is a function with respect to (119906120572) only and 1198652satisfies

in (126)(ii) If 119865

1and 119865

2are locally dually flat then 119865 is locally

dually flat if and only if 119891 is a function with respect(119906

120572) only and 119865

2satisfies in (128)

ByTheorem 29 we conclude the following

Corollary 30 There is not exist any locally dually flat propertwisted product Finsler manifold

References

[1] J K Beem P E Ehrlich and K L Easley Global LorentzianGeometry vol 202 of Monographs and Textbooks in Pure andApplied Mathematics Marcel Dekker New York NY USA 2ndedition 1996

[2] G Ganchev and V Mihova ldquoRiemannian manifolds of quasi-constant sectional curvaturesrdquo Journal fur die Reine und Ange-wandte Mathematik vol 522 pp 119ndash141 2000

[3] B OrsquoNeill Semi-Riemannian Geometry With Applications toRelativity vol 103 of Pure and Applied Mathematics AcademicPress New York NY USA 1983

[4] M Lohnherr and H Reckziegel ldquoOn ruled real hypersurfacesin complex space formsrdquo Geometriae Dedicata vol 74 no 3pp 267ndash286 1999

[5] B-Y Chen F Dillen L Verstraelen and L VranckenldquoLagrangian isometric immersions of a real-space-form119872119899

(119888)

into a complex-space-form119872119899(4119888)rdquoMathematical Proceedings

of the Cambridge Philosophical Society vol 124 no 1 pp 107ndash125 1998

[6] N Koike ldquoThe decomposition of curvature netted hypersur-facesrdquo Geometriae Dedicata vol 54 no 1 pp 1ndash11 1995

[7] B-y Chen Geometry of Submanifolds and its ApplicationsScience University of Tokyo Tokyo Japan 1981

[8] R Ponge and H Reckziegel ldquoTwisted products in pseudo-Riemannian geometryrdquo Geometriae Dedicata vol 48 no 1 pp15ndash25 1993

[9] B-Y Chen ldquoTwisted product CR-submanifolds in Kaehlermanifoldsrdquo Tamsui Oxford Journal of Mathematical Sciencesvol 16 no 2 pp 105ndash121 2000

[10] L Kozma I R Peter and H Shimada ldquoOn the twisted productof Finsler manifoldsrdquo Reports on Mathematical Physics vol 57no 3 pp 375ndash383 2006

[11] M Matsumoto and C Shibata ldquoOn semi-C-reducibility T-tensor= 0 and S4-likeness of Finsler spacesrdquo Journal of Math-ematics of Kyoto University vol 19 no 2 pp 301ndash314 1979

[12] B Najafi Z Shen and A Tayebi ldquoFinsler metrics of scalar flagcurvature with special non-Riemannian curvature propertiesrdquoGeometriae Dedicata vol 131 pp 87ndash97 2008

[13] E Peyghan andAHeydari ldquoConformal vector fields on tangentbundle of a Riemannian manifoldrdquo Journal of MathematicalAnalysis and Applications vol 347 no 1 pp 136ndash142 2008

[14] Z Shen Differential Geometry of Spray and Finsler SpacesKluwer Academic Publishers Dordrecht The Netherlands2001

[15] M Matsumoto and S-I Hojo ldquoA conclusive theorem on C-reducible Finsler spacesrdquo The Tensor Society vol 32 no 2 pp225ndash230 1978

[16] D Bao S-S Chern and Z Shen An Introduction to Riemann-Finsler Geometry vol 200 of Graduate Texts in MathematicsSpringer New York NY USA 2000

[17] Z Shen ldquoRiemann-Finsler geometry with applications to infor-mation geometryrdquo Chinese Annals of Mathematics B vol 27 no1 pp 73ndash94 2006

[18] Y Ichijyo ldquoFinsler manifolds modeled on a Minkowski spacerdquoJournal of Mathematics of Kyoto University vol 16 no 3 pp639ndash652 1976

[19] X Chen and Z Shen ldquoOn Douglas metricsrdquo PublicationesMathematicae Debrecen vol 66 no 3-4 pp 503ndash512 2005

[20] L Berwald ldquoUber Parallelubertragung in Raumen mit allge-meiner Massbestimmungrdquo Jahresbericht der DMVmdashDeutscheMathematiker-Vereinigung vol 34 pp 213ndash220 1926

[21] M Matsumoto ldquoOn Finsler spaces with Randersrsquo metric andspecial forms of important tensorsrdquo Journal of Mathematics ofKyoto University vol 14 pp 477ndash498 1974

12 ISRN Geometry

[22] A Tayebi and E Peyghan ldquoOn special Berwald metricsrdquoSymmetry Integrability and Geometry vol 6 article 008 9pages 2010

[23] S-I Amari and H NagaokaMethods of Information Geometryvol 191 of Translations of Mathematical Monographs AmericanMathematical Society Providence RI USA 2000

[24] X Cheng Z Shen and Y Zhou ldquoOn locally dually flat Finslermetricsrdquo International Journal ofMathematics vol 21 no 11 pp1531ndash1543 2010

[25] Q Xia ldquoOn a class of locally dually flat Finsler metrics ofisotropic flag curvaturerdquo Publicationes Mathematicae Debrecenvol 78 no 1 pp 169ndash190 2011

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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

8 ISRN Geometry

R 119894

120572 120573119897=120575119905

120575119905119906120573(G119903

120572119862119894

119897119903) minus120575119905

120575119905119909119897119891 (119891

119894119892120572120573minus 119891G120582

ℎ119892119894ℎ119862120572120573120582)

minus G119898

120573G119904

120572119862119894

119903119898119862119903

119897119904+ (119891

119894119892120583120573minus 119891G120582

ℎ119892119894ℎ119862120583120573120582)

times (119891119897120575120583

120572minus 119891G]

119897119862120583

120572]) minus 119891119892119903ℎ(119865

119894

119903119897minus119872

119894

119903119897)

times (119891ℎ119892120572120573minus119891G120582

ℎ119862120572120573120582)minusG119903

120583119862119894

119897119903(119865

120583

120572120573+ 119873

120583

120572120573minus119872

120583

120572120573)

(86)

R 119894

119895 120573119897= minus

120575119905

120575119905119909119897(G119903

120573119862119894

119895119903) minus

120575119905

120575119905119906120573(119865

119894

119895119897minus119872

119894

119895119897)

minus (119865119894

119897119903minus119872

119894

119897119903)G119904

120573119862119903

119895119904minus 119891

minus1G119903

120572119862119894

119897119903

times (119891119895120575120572

120573minus 119891G120583

119895119862120572

120573120583) + G119904

120573119862119894

119903119904(119865

119903

119895119897minus119872

119903

119895119897)

+ 119891minus1G119903

120583119862119895119897119903(119891

119894120575120583

120573minus 119891G120582

ℎ119892119894ℎ119862120583

120573120582)

(87)

R 119894

120572 120573120582= minus

120575119905

120575119905119906120582(119891119891

119894119892120572120573minus 119891

2119892119894ℎG120583

ℎ119862120572120573120583) + 119891G119904

120582119862119894

119903119904

times(119891119903119892120572120573minus119891G120583

119897119862120572120573120583119892119903119897)minus119891 (119891

119894119892120582120583minus119891119892

119894ℎG120581

ℎ119862120582120583120581)

times (119865120583

120572120573+ 119873

120583

120572120573minus119872

120583

120572120573) minus C

120573

120582

(88)

R 120574

119895 119896119897=

120575119905

120575119905119909119897(119891

minus2119892120574120582G119903

120582119862119895119896119903) + 119891

minus2119892120574120582G119904

120582119862119897119903119904

times (119865119903

119895119896minus119872

119903

119895119896) + 119891

minus3G119903

120583119862119895119896119903(119891

119897119892120574120583minus 119891G120572

119897119862120574120583

120572)

minus C119896

119897

(89)

R 120574

119895 120573119897=120575119905

120575119905119909119897(119891

minus1119891119895120575120574

120573minus 119891G120572

119895119862120574

120572120573)

minus120575119905

120575119905119906120573(119891

minus2119892120574120582G119903

120582119862119895119897119903) minus 119891

minus2119892120574120582G119904

120582G119898

120573119862ℎ

119897119904119862ℎ119895119898

+ 119891minus2(119891

119897120575120574

120583minus 119891G120572

119897119862120574

120583120572) (119891

119895120575120583

120573minus 119891G]

119895119862120583

120573])

minus 119891minus1(119891

119903120575120574

120573minus 119891G120572

119903119862120574

120573120572) (119865

119903

119895119897minus119872

119903

119895119897)

minus 119891minus2119892120583120582G119903

120582119862119895119897119903(119865

120574

120573120583+ 119873

120574

120573120583minus119872

120574

120573120583)

(90)

R 120574

120572 120573119897=120575119905

120575119905119909119897(119865

120574

120572120573+ 119873

120574

120572120573minus119872

120574

120572120573)

minus120575119905

120575119905119906120573(119891

minus1119891119897120575120574

120572minus G120583

119897119862120574

120572120583)

minus 119891minus1119892120574120582G119904

120582119862ℎ

119897119904(119891

ℎ119892120572120573minus 119891G120583

ℎ119862120572120573120583)

+ 119891minus1(119891

119897120575120574

120583minus 119891G120581

119897119862120574

120583120581) (119865

120583

120572120573+ 119873

120583

120572120573minus119872

120583

120572120573)

+ 119891minus1G119904

120572119862119903

119897119904(119891

119903120575120574

120573minus 119891G120581

119903119862120574

120573120581)

minus 119891minus1(119891

119897120575120583

120572minus 119891G120581

119897119862120583

120572120581) (119865

120574

120573120583+ 119873

120574

120573120583minus119872

120574

120573120583)

(91)

R 120574

119895 120573120582=

120575119905

120575119905119906120582(119891

minus1119891119895120575120574

120573minus G120572

119895119862120574

120572120573)

+ 119891minus1(119865

120574

120572120582+ 119873

120574

120572120582minus119872

120574

120572120582) (119891

119895120575120572

120573minus 119891G]

119895119862120572

120573])

minus119891minus1G119898

120573119862119903

119895119898(119891

119903120575120574

120582minus 119891G120572

119903119862120574

120582120572) minus C

120573

120582

(92)

R 120574

120572 119896119897=

120575119905

120575119905119909119897(119891

minus1119891119896120575120574

120572minus G120583

119896119862120574

120572120583)

+ 119891minus2(119891

119897120575120574

120573minus 119891G120581

119897119862120574

120573120581) (119891

119896120575120573

120572minus 119891G]

119896119862120573

120572])

minus119891minus2G119904

120583G119898

120572119892120574120583119862ℎ

119897119904119862ℎ119896119898 minus C

119896

119897

(93)

R 120574

120572 120573120582= 119877

120574

120572 120573120582minus 119872

120581

120582

120597119865120574

120572120573

120597V120581+

120575119905119872

120574

120572120573

120575119905119906120582+119865

120574

120582120583119872

120583

120572120573+119872

120574

120582120583119865120583

120572120573

minus119872120574

120582120583119872

120583

120572120573+120575119905119873

120574

120572120582

120575119905119906120573+ 119865

120574

120573120583119873

120583

120572120582

+ 119873120574

120573120583119865120583

120572120582+ 119873

120574

120573120583119873

120583

120572120582+ 119873

120574

120582120583119872

120583

120572120573

+ 119873120583

120572120573119872

120574

120582120583+ (119892

119903119904120575120574

120582119891119904minus 119891119892

119903119904G120581

119904119862120574

120582120581)

times (119892120572120573119891119903minus 119891G120583

119903119862120572120573120583) minusC

120573

120582

(94)

where

119872119894

119895119896= 119872

119903

119896119862119894

119895119903+119872

119903

119895119862119894

119896119903minus119872

119903

ℎ119892119894ℎ119862119895119896119903

119872120574

120572120573= 119872

120583

120573119862120574

120572120583+119872

120583

120572119862120574

120573120583minus119872

120583

] 119892120574]119862120572120573120583

119873120574

120572120573= 119891

minus1(119891

120573120575120574

120572+ 119891

120572120575120574

120573minus 119891

120582119892120574120582119892120572120573)

(95)

and C119894

119895denotes the interchange of indices 119894 119895 and subtraction

ByTheorem 18 we have the following

Theorem 16 Let (1198721times119891119872

2 119865) be a flat twisted product

Finsler manifold and let (1198721 119865

1) be Riemannian If 119891 is a

function on1198722 only then (119872

1 119865

1) is locally flat

Similarly we get the following

Theorem 17 Let (1198721times119891119872

2 119865) be a flat twisted product

Finsler manifold and let (1198722 119865

2) be Riemannian If 119891 is a

function on 1198721 only then (119872

2 119865

2) is a space of positive

constant curvature ||119892119903119886119889119891||2

ISRN Geometry 9

Proof Since 1198722is Riemannain and 119891 is a function on 119872

1

then by (94) we obtain

R 120574

120572 120573120582= 119877

120574

120572 120573120582+1003817100381710038171003817119892119903119886119889119891

1003817100381710038171003817

2

(120575120574

120582119892120572120573minus 120575

120574

120573119892120572120582) (96)

Since (1198721times119891119872

2 119865) is flat then R120574

120572 120573120582= 0 Thus the proof is

complete

Theorem 18 Let (1198721times119891119872

2 119865) be a twisted product Rieman-

nian manifold and let 119891 be a function on 1198722 only Then

(1198721times119891119872

2 119865) is flat if and only if (119872

1 119865

1) is flat and the

Riemannian curvature of (1198722 119865

2) satisfies in the following

equation

119877120574

120572 120573120582= 120575119905119873

120574

120572120582

120575119905119906120573+ 119865

120574

120573120583119873

120583

120572120582+ 119873

120574

120573120583119865120583

120572120582+ 119873

120574

120573120583119873

120583

120572120582 minus C

120573

120582

(97)

5 Twisted Product Finsler Manifolds withNon-Riemannian Curvature Properties

There are several important non-Riemannian quantities suchas the Berwald curvature B the mean Berwald curvatureE and the Landsberg curvature L [22] They all vanishfor Riemannian metrics hence they are said to be non-Riemannian In this section we find some necessary and suf-ficient conditions underwhich a twisted product Riemannianmanifold are Berwaldian of isotropic Berwald curvatureof isotropic mean Berwald curvature First we prove thefollowing

Lemma 19 Let (1198721times119891119872

2 119865) be a twisted product Finsler

manifold Then the coefficients of Berwald curvature are asfollows

B120574

120572120573120582= 119861

120574

120572120573120582+ 119891

minus1(119862

120574]120582120572120573119891]119865

2

2+ 2119862

120574]120572120573119891]V120582

+ 2119862120574]120572120582119891]V120573 + 2119862

120574]120572119891]119892120582120573

+ 2119862120574]120582120573119891]V120572 + 2119862

120574]120573119891]119892120582120572

+ 2119862120574]120582119891]119892120572120573 minus 2119862120572120573120582119891

120574)

(98)

B119896

119894119895119897= 119861

119896

119894119895119897+ 119891119862

119896ℎ

119897119895119894119891ℎ1198652

2 (99)

B119896

119894120573119897= 2119891119862

119896ℎ

119894119897119891ℎV120573 (100)

B119896

120572120573119897= 2119891119892

120572120573119862119896ℎ

119897119891ℎ (101)

B119896

120572120573120582= minus 2119891119862

120572120573120582119891119896 (102)

B120574

119894120573120582= B120574

119894119895120582= B120574

119894119895119896= 0 (103)

Let (1198721times119891119872

2 119865) is a Berwald manifold Then we have

B119889

119886119887119888= 0 By using (102) we get

119862120572120573120582119891119896= 0 (104)

Multiplying this equation in 119892119896119903 we obtain

119862120572120573120582119891119903= 0 (105)

Thus if 119891 is not constant on1198721 then we have 119862

120572120573120582= 0 Also

from (101) we result that

119862119896ℎ

119897119891ℎ= 0 (106)

Differentiating this equation with respect to 119910119895 gives us

119862119896ℎ

119897119895119891ℎ= 0 (107)

Similarly we obtain

119862119896ℎ

119897119895119894119891ℎ= 0 (108)

Setting the last equation in (99) implies that 119861119896119894119895119897= 0 that

is (1198721 119865

1) is Berwaldian These explanations give us the

following theorem

Theorem 20 Let (1198721times119891119872

2 119865) be a twisted product Finsler

manifold and let119891 be not constant on1198721Then (119872

1times119891119872

2 119865)

is Berwaldian if and only if (1198721 119865

1) is Berwaldian (119872

2 119865

2) is

Riemannian and the equation 119862119896ℎ119897119891ℎ= 0 is hold

But if 119891 is constant on1198721 that is 119891

119894= 0 then we get the

following

Theorem 21 Let (1198721times119891119872

2 119865) be a twisted product Finsler

manifold and 119891 is constant on 1198721 Then (119872

1times119891119872

2 119865)

is Berwaldian if and only if (1198721 119865

1) is Berwaldian and

the Berwald curvature of (1198722 119865

2) satisfies in the following

equation

119861120574

120572120573120582= minus 119891

minus1(119862

120574]120573120572120582119891]119865

2

2+ 2119862

120574]120573120572119891]V120582 + 2119862

120574]120582120572119891]V120573

+ 2119862120574]120572119891]119892120582120573 + 2119862

120574]120573120582119891]V120572

+ 2119862120574]120573119891]119892120582120572 + 2119862

120574]120582119891]119892120572120573

minus 2119892120574]119862120572120573120582119891])

(109)

Here we consider twisted product Finsler manifold(119872

1times119891119872

2 119865) of isotropic Berwald curvature

Theorem 22 Every isotropic Berwald twisted product Finslermanifold (119872

1times119891119872

2 119865) is a Berwald manifold

Proof Let (1198721times119891119872

2 119865) be an isotropic Berwald manifold

Then we have

B119889

119886119887119888= 119888119865

minus1h119889

119886h119887119888+ h119889

119887h119886119888+ h119889

119888h119886119887+ 2C

119886119887119888y119889 (110)

where 119888 = 119888(x) is a function on119872 Setting 119886 = 119895 119887 = 119896 119888 = 119897and 119889 = 120574 and using (103) imply that

119888119865minus13

1198652119910119895119910119896119910119897V120574 minus V120574 (119910

119895119892119896119897+ 119910

119896119892119895119897+ 119910

119897119892119895119896) = 0 (111)

Multiplying the previous equation in 119910119895119910119896 we derive that119888119891

21198652

11198652

2= 0 Thus we have 119888 = 0 that is (119872

1times119891119872

2) is

Berwaldian

10 ISRN Geometry

Now we are going to study twisted product Finslermanifold of isotropicmean Berwald curvature For this workwe must compute the coefficients of mean Berwald curvatureof a twisted product Finsler manifold

Lemma 23 Let (1198721times119891119872

2 119865) be a twisted product Finsler

manifold Then the coefficients of mean Berwald curvature areas follows

E120572120573= 119864

120572120573+ 119891119892

120572120573119868ℎ119891ℎ+1

2119891119868

]120572120573119891]119865

2

2+ 119891

minus1119891]

times (119862120574]120572120573

V120574+ 119868

]120572V120573+ 119868

]120573V120572+ 119862

]120572120573+ 119868

]119892120572120573)

(112)

E119894119895= 119864

119894119895+1

2119891119868

ℎ

119895119894119891ℎ1198652

2 (113)

E119894120573= 119891119868

ℎ

119894119891ℎV120573 (114)

where 119864119894119895

and 119864120572120573

are the coefficients of mean Berwaldcurvature of (119872

1 119865

1) and (119872

2 119865

2) respectively

Proof By definition and Lemma 19 we get the proof

Theorem24 The twisted product Finslermanifold (1198721times119891119872

2

119865) is weakly Berwald if and only if (1198721 119865

1) is weakly Berwald

119868ℎ119891ℎ= 0 and the following hold

119864120572120573= minus

1

2119891119868

]120572120573119891]119865

2

2minus 119891

minus1119891]

times (119862120574]120572120573

V120574+ 119868

]120572V120573+ 119868

]120573V120572+ 119862

]120572120573+ 119868

]119892120572120573)

(115)

Proof If (1198721times119891119872

2) be a weakly Berwald manifold then we

have

E120572120573= E

119894119895= E

119894120573= 0 (116)

Thus by using (114) we result that 119868ℎ119894119891ℎ= 0 This equation

implies that

119868ℎ

119895119894119891ℎ= 0 119868

ℎ119891ℎ= 0 (117)

By setting these equations in (112) and (113) we conclude that119864119894119895= 0 and 119864

120572120573satisfies in (115)

Now if 119891 is constant on1198722 then (115) implies that 119864

120572120573=

0 Thus we conclude the following

Corollary 25 Let (1198721times119891119872

2 119865) be a twisted product Finsler

manifold and let 119891 be a function on 1198721 only Then

(1198721times119891119872

2 119865) is weakly Berwald if and only if (119872

1 119865

1) and

(1198722 119865

2) are weakly Berwald manifolds and 119868ℎ119891

ℎ= 0

Now we consider twisted product Finsler manifolds withisotropic mean Berwald curvature It is remarkable that as aconsequence of Lemma 23 we have the following

Lemma26 Twisted product Finslermanifold (1198721times119891119872

2 119865) is

isotropic mean Berwald manifold if and only if

119864120572120573+ 119891119892

120572120573119868ℎ119891ℎ+1

2119891119868

]120572120573119891]119865

2

2+ 119891

minus1119891]

times (119862120574]120572120573

V120574+ 119868

]120572V120573+ 119868

]120573V120572+ 119862

]120572120573+ 119868

]119892120572120573)

minus119899 + 1

2119888119891

2119865minus1(119892

120572120573minus1198912

1198652V120572V120573) = 0

(118)

119864119894119895+1

2119891119868

ℎ

119895119894119891ℎ1198652

2minus119899 + 1

2119888119865

minus1(119892

119894119895minus1

1198652119910119894119910119895) = 0 (119)

119888 (119899 + 1) 119865minus3119910119894+ 119891119868

ℎ

119894119891ℎ= 0 (120)

where 119888 = 119888(x) is a scalar function on119872

Theorem 27 Every twisted product Finsler manifold(119872

1times119891119872

2 119865) with isotropic mean Berwald curvature is a

weakly Berwald manifold

Proof Suppose that 119865 is isotropic mean Berwald twistedproduct Finsler metric Then differentiating (120) withrespect to V120574 gives us

119888 (119899 + 1) 1198912119865minus5V

120574119910119894= 0 (121)

Thus we conclude that 119888 = 0 This implies that 119865 reduces to aweakly Berwald metric

6 Locally Dually Flat Twisted ProductFinsler Manifolds

In [23] Amari and Nagaoka introduced the notion of duallyflat Riemannian metrics when they study the informationgeometry on Riemannian manifolds Information geometryhas emerged from investigating the geometrical structure ofa family of probability distributions and has been appliedsuccessfully to various areas including statistical inferencecontrol system theory andmultiterminal information theoryIn Finsler geometry Shen extends the notion of locally duallyflatness for Finsler metrics [17] Dually flat Finsler metricsform a special and valuable class of Finsler metrics in Finslerinformation geometry which play a very important role instudying flat Finsler information structure [24 25]

In this section we study locally dually flat twisted productFinsler metrics It is remarkable that a Finsler metric 119865 =119865(x y) on a manifold119872 is said to be locally dually flat if atany point there is a standard coordinate system (x119886 y119886) in119879119872such that it satisfies

12059721198652

120597x119887120597y119886y119887 = 2120597119865

2

120597x119886 (122)

In this case the coordinate (x119886) is called an adapted localcoordinate system Byusing (122) we can obtain the followinglemma

ISRN Geometry 11

Lemma 28 Let (1198721times119891119872

2 119865) be a twisted product Finsler

manifold Then 119865 is locally dually flat if and only if 1198651and 119865

2

satisfy in the following equations

12059721198652

1

120597119909119896120597119910119897119910119896= 2120597119865

2

1

120597119909119897+ 4119891119891

1198971198652

2 (123)

4119891119896V120573119910119896+ 119891

12059721198652

2

120597119906120572120597V120573V120572 + 4119891

120572V120573V120572 = 2119891

1205971198652

2

120597119906120573+ 4119891

1205731198652

2

(124)

Now let 119865 be a locally dually flat Finsler metric Takingderivative with respect to V120574 from (123) yields 119891

119897= 0 which

means that 119891 is a constant function on1198721 In this case the

relations (123) and (124) reduce to the following

12059721198652

1

120597119909119896120597119910119897119910119896= 2120597119865

2

1

120597119909119897 (125)

11989112059721198652

2

120597119906120572120597V120573V120572 + 4119891

120572V120573V120572 = 2119891

1205971198652

2

120597119906120573+ 4119891

1205731198652

2 (126)

By (125) we deduce that 1198651is locally dually flat

Now we assume that 1198651and 119865

2are locally dually flat

Finsler metrics Then we have

12059721198652

1

120597119909119896120597119910119897119910119896= 2120597119865

2

1

120597119909119897

12059721198652

2

120597119906120572120597V120573V120572 = 2

1205971198652

2

120597119906120573

(127)

By (127) we derive that (123) and (124) are hold if and only ifthe following hold

119891119897= 0 119891

120572V120573V120572 = 119891

1205731198652

2 (128)

Therefore we can conclude the following

Theorem 29 Let (1198721times119891119872

2 119865) be a twisted product Finsler

manifold

(i) If 119865 is locally dually flat then 1198651is locally dually flat 119891

is a function with respect to (119906120572) only and 1198652satisfies

in (126)(ii) If 119865

1and 119865

2are locally dually flat then 119865 is locally

dually flat if and only if 119891 is a function with respect(119906

120572) only and 119865

2satisfies in (128)

ByTheorem 29 we conclude the following

Corollary 30 There is not exist any locally dually flat propertwisted product Finsler manifold

References

[1] J K Beem P E Ehrlich and K L Easley Global LorentzianGeometry vol 202 of Monographs and Textbooks in Pure andApplied Mathematics Marcel Dekker New York NY USA 2ndedition 1996

[2] G Ganchev and V Mihova ldquoRiemannian manifolds of quasi-constant sectional curvaturesrdquo Journal fur die Reine und Ange-wandte Mathematik vol 522 pp 119ndash141 2000

[3] B OrsquoNeill Semi-Riemannian Geometry With Applications toRelativity vol 103 of Pure and Applied Mathematics AcademicPress New York NY USA 1983

[4] M Lohnherr and H Reckziegel ldquoOn ruled real hypersurfacesin complex space formsrdquo Geometriae Dedicata vol 74 no 3pp 267ndash286 1999

[5] B-Y Chen F Dillen L Verstraelen and L VranckenldquoLagrangian isometric immersions of a real-space-form119872119899

(119888)

into a complex-space-form119872119899(4119888)rdquoMathematical Proceedings

of the Cambridge Philosophical Society vol 124 no 1 pp 107ndash125 1998

[6] N Koike ldquoThe decomposition of curvature netted hypersur-facesrdquo Geometriae Dedicata vol 54 no 1 pp 1ndash11 1995

[7] B-y Chen Geometry of Submanifolds and its ApplicationsScience University of Tokyo Tokyo Japan 1981

[8] R Ponge and H Reckziegel ldquoTwisted products in pseudo-Riemannian geometryrdquo Geometriae Dedicata vol 48 no 1 pp15ndash25 1993

[9] B-Y Chen ldquoTwisted product CR-submanifolds in Kaehlermanifoldsrdquo Tamsui Oxford Journal of Mathematical Sciencesvol 16 no 2 pp 105ndash121 2000

[10] L Kozma I R Peter and H Shimada ldquoOn the twisted productof Finsler manifoldsrdquo Reports on Mathematical Physics vol 57no 3 pp 375ndash383 2006

[11] M Matsumoto and C Shibata ldquoOn semi-C-reducibility T-tensor= 0 and S4-likeness of Finsler spacesrdquo Journal of Math-ematics of Kyoto University vol 19 no 2 pp 301ndash314 1979

[12] B Najafi Z Shen and A Tayebi ldquoFinsler metrics of scalar flagcurvature with special non-Riemannian curvature propertiesrdquoGeometriae Dedicata vol 131 pp 87ndash97 2008

[13] E Peyghan andAHeydari ldquoConformal vector fields on tangentbundle of a Riemannian manifoldrdquo Journal of MathematicalAnalysis and Applications vol 347 no 1 pp 136ndash142 2008

[14] Z Shen Differential Geometry of Spray and Finsler SpacesKluwer Academic Publishers Dordrecht The Netherlands2001

[15] M Matsumoto and S-I Hojo ldquoA conclusive theorem on C-reducible Finsler spacesrdquo The Tensor Society vol 32 no 2 pp225ndash230 1978

[16] D Bao S-S Chern and Z Shen An Introduction to Riemann-Finsler Geometry vol 200 of Graduate Texts in MathematicsSpringer New York NY USA 2000

[17] Z Shen ldquoRiemann-Finsler geometry with applications to infor-mation geometryrdquo Chinese Annals of Mathematics B vol 27 no1 pp 73ndash94 2006

[18] Y Ichijyo ldquoFinsler manifolds modeled on a Minkowski spacerdquoJournal of Mathematics of Kyoto University vol 16 no 3 pp639ndash652 1976

[19] X Chen and Z Shen ldquoOn Douglas metricsrdquo PublicationesMathematicae Debrecen vol 66 no 3-4 pp 503ndash512 2005

[20] L Berwald ldquoUber Parallelubertragung in Raumen mit allge-meiner Massbestimmungrdquo Jahresbericht der DMVmdashDeutscheMathematiker-Vereinigung vol 34 pp 213ndash220 1926

[21] M Matsumoto ldquoOn Finsler spaces with Randersrsquo metric andspecial forms of important tensorsrdquo Journal of Mathematics ofKyoto University vol 14 pp 477ndash498 1974

12 ISRN Geometry

[22] A Tayebi and E Peyghan ldquoOn special Berwald metricsrdquoSymmetry Integrability and Geometry vol 6 article 008 9pages 2010

[23] S-I Amari and H NagaokaMethods of Information Geometryvol 191 of Translations of Mathematical Monographs AmericanMathematical Society Providence RI USA 2000

[24] X Cheng Z Shen and Y Zhou ldquoOn locally dually flat Finslermetricsrdquo International Journal ofMathematics vol 21 no 11 pp1531ndash1543 2010

[25] Q Xia ldquoOn a class of locally dually flat Finsler metrics ofisotropic flag curvaturerdquo Publicationes Mathematicae Debrecenvol 78 no 1 pp 169ndash190 2011

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

ISRN Geometry 9

Proof Since 1198722is Riemannain and 119891 is a function on 119872

1

then by (94) we obtain

R 120574

120572 120573120582= 119877

120574

120572 120573120582+1003817100381710038171003817119892119903119886119889119891

1003817100381710038171003817

2

(120575120574

120582119892120572120573minus 120575

120574

120573119892120572120582) (96)

Since (1198721times119891119872

2 119865) is flat then R120574

120572 120573120582= 0 Thus the proof is

complete

Theorem 18 Let (1198721times119891119872

2 119865) be a twisted product Rieman-

nian manifold and let 119891 be a function on 1198722 only Then

(1198721times119891119872

2 119865) is flat if and only if (119872

1 119865

1) is flat and the

Riemannian curvature of (1198722 119865

2) satisfies in the following

equation

119877120574

120572 120573120582= 120575119905119873

120574

120572120582

120575119905119906120573+ 119865

120574

120573120583119873

120583

120572120582+ 119873

120574

120573120583119865120583

120572120582+ 119873

120574

120573120583119873

120583

120572120582 minus C

120573

120582

(97)

5 Twisted Product Finsler Manifolds withNon-Riemannian Curvature Properties

There are several important non-Riemannian quantities suchas the Berwald curvature B the mean Berwald curvatureE and the Landsberg curvature L [22] They all vanishfor Riemannian metrics hence they are said to be non-Riemannian In this section we find some necessary and suf-ficient conditions underwhich a twisted product Riemannianmanifold are Berwaldian of isotropic Berwald curvatureof isotropic mean Berwald curvature First we prove thefollowing

Lemma 19 Let (1198721times119891119872

2 119865) be a twisted product Finsler

manifold Then the coefficients of Berwald curvature are asfollows

B120574

120572120573120582= 119861

120574

120572120573120582+ 119891

minus1(119862

120574]120582120572120573119891]119865

2

2+ 2119862

120574]120572120573119891]V120582

+ 2119862120574]120572120582119891]V120573 + 2119862

120574]120572119891]119892120582120573

+ 2119862120574]120582120573119891]V120572 + 2119862

120574]120573119891]119892120582120572

+ 2119862120574]120582119891]119892120572120573 minus 2119862120572120573120582119891

120574)

(98)

B119896

119894119895119897= 119861

119896

119894119895119897+ 119891119862

119896ℎ

119897119895119894119891ℎ1198652

2 (99)

B119896

119894120573119897= 2119891119862

119896ℎ

119894119897119891ℎV120573 (100)

B119896

120572120573119897= 2119891119892

120572120573119862119896ℎ

119897119891ℎ (101)

B119896

120572120573120582= minus 2119891119862

120572120573120582119891119896 (102)

B120574

119894120573120582= B120574

119894119895120582= B120574

119894119895119896= 0 (103)

Let (1198721times119891119872

2 119865) is a Berwald manifold Then we have

B119889

119886119887119888= 0 By using (102) we get

119862120572120573120582119891119896= 0 (104)

Multiplying this equation in 119892119896119903 we obtain

119862120572120573120582119891119903= 0 (105)

Thus if 119891 is not constant on1198721 then we have 119862

120572120573120582= 0 Also

from (101) we result that

119862119896ℎ

119897119891ℎ= 0 (106)

Differentiating this equation with respect to 119910119895 gives us

119862119896ℎ

119897119895119891ℎ= 0 (107)

Similarly we obtain

119862119896ℎ

119897119895119894119891ℎ= 0 (108)

Setting the last equation in (99) implies that 119861119896119894119895119897= 0 that

is (1198721 119865

1) is Berwaldian These explanations give us the

following theorem

Theorem 20 Let (1198721times119891119872

2 119865) be a twisted product Finsler

manifold and let119891 be not constant on1198721Then (119872

1times119891119872

2 119865)

is Berwaldian if and only if (1198721 119865

1) is Berwaldian (119872

2 119865

2) is

Riemannian and the equation 119862119896ℎ119897119891ℎ= 0 is hold

But if 119891 is constant on1198721 that is 119891

119894= 0 then we get the

following

Theorem 21 Let (1198721times119891119872

2 119865) be a twisted product Finsler

manifold and 119891 is constant on 1198721 Then (119872

1times119891119872

2 119865)

is Berwaldian if and only if (1198721 119865

1) is Berwaldian and

the Berwald curvature of (1198722 119865

2) satisfies in the following

equation

119861120574

120572120573120582= minus 119891

minus1(119862

120574]120573120572120582119891]119865

2

2+ 2119862

120574]120573120572119891]V120582 + 2119862

120574]120582120572119891]V120573

+ 2119862120574]120572119891]119892120582120573 + 2119862

120574]120573120582119891]V120572

+ 2119862120574]120573119891]119892120582120572 + 2119862

120574]120582119891]119892120572120573

minus 2119892120574]119862120572120573120582119891])

(109)

Here we consider twisted product Finsler manifold(119872

1times119891119872

2 119865) of isotropic Berwald curvature

Theorem 22 Every isotropic Berwald twisted product Finslermanifold (119872

1times119891119872

2 119865) is a Berwald manifold

Proof Let (1198721times119891119872

2 119865) be an isotropic Berwald manifold

Then we have

B119889

119886119887119888= 119888119865

minus1h119889

119886h119887119888+ h119889

119887h119886119888+ h119889

119888h119886119887+ 2C

119886119887119888y119889 (110)

where 119888 = 119888(x) is a function on119872 Setting 119886 = 119895 119887 = 119896 119888 = 119897and 119889 = 120574 and using (103) imply that

119888119865minus13

1198652119910119895119910119896119910119897V120574 minus V120574 (119910

119895119892119896119897+ 119910

119896119892119895119897+ 119910

119897119892119895119896) = 0 (111)

Multiplying the previous equation in 119910119895119910119896 we derive that119888119891

21198652

11198652

2= 0 Thus we have 119888 = 0 that is (119872

1times119891119872

2) is

Berwaldian

10 ISRN Geometry

Now we are going to study twisted product Finslermanifold of isotropicmean Berwald curvature For this workwe must compute the coefficients of mean Berwald curvatureof a twisted product Finsler manifold

Lemma 23 Let (1198721times119891119872

2 119865) be a twisted product Finsler

manifold Then the coefficients of mean Berwald curvature areas follows

E120572120573= 119864

120572120573+ 119891119892

120572120573119868ℎ119891ℎ+1

2119891119868

]120572120573119891]119865

2

2+ 119891

minus1119891]

times (119862120574]120572120573

V120574+ 119868

]120572V120573+ 119868

]120573V120572+ 119862

]120572120573+ 119868

]119892120572120573)

(112)

E119894119895= 119864

119894119895+1

2119891119868

ℎ

119895119894119891ℎ1198652

2 (113)

E119894120573= 119891119868

ℎ

119894119891ℎV120573 (114)

where 119864119894119895

and 119864120572120573

are the coefficients of mean Berwaldcurvature of (119872

1 119865

1) and (119872

2 119865

2) respectively

Proof By definition and Lemma 19 we get the proof

Theorem24 The twisted product Finslermanifold (1198721times119891119872

2

119865) is weakly Berwald if and only if (1198721 119865

1) is weakly Berwald

119868ℎ119891ℎ= 0 and the following hold

119864120572120573= minus

1

2119891119868

]120572120573119891]119865

2

2minus 119891

minus1119891]

times (119862120574]120572120573

V120574+ 119868

]120572V120573+ 119868

]120573V120572+ 119862

]120572120573+ 119868

]119892120572120573)

(115)

Proof If (1198721times119891119872

2) be a weakly Berwald manifold then we

have

E120572120573= E

119894119895= E

119894120573= 0 (116)

Thus by using (114) we result that 119868ℎ119894119891ℎ= 0 This equation

implies that

119868ℎ

119895119894119891ℎ= 0 119868

ℎ119891ℎ= 0 (117)

By setting these equations in (112) and (113) we conclude that119864119894119895= 0 and 119864

120572120573satisfies in (115)

Now if 119891 is constant on1198722 then (115) implies that 119864

120572120573=

0 Thus we conclude the following

Corollary 25 Let (1198721times119891119872

2 119865) be a twisted product Finsler

manifold and let 119891 be a function on 1198721 only Then

(1198721times119891119872

2 119865) is weakly Berwald if and only if (119872

1 119865

1) and

(1198722 119865

2) are weakly Berwald manifolds and 119868ℎ119891

ℎ= 0

Now we consider twisted product Finsler manifolds withisotropic mean Berwald curvature It is remarkable that as aconsequence of Lemma 23 we have the following

Lemma26 Twisted product Finslermanifold (1198721times119891119872

2 119865) is

isotropic mean Berwald manifold if and only if

119864120572120573+ 119891119892

120572120573119868ℎ119891ℎ+1

2119891119868

]120572120573119891]119865

2

2+ 119891

minus1119891]

times (119862120574]120572120573

V120574+ 119868

]120572V120573+ 119868

]120573V120572+ 119862

]120572120573+ 119868

]119892120572120573)

minus119899 + 1

2119888119891

2119865minus1(119892

120572120573minus1198912

1198652V120572V120573) = 0

(118)

119864119894119895+1

2119891119868

ℎ

119895119894119891ℎ1198652

2minus119899 + 1

2119888119865

minus1(119892

119894119895minus1

1198652119910119894119910119895) = 0 (119)

119888 (119899 + 1) 119865minus3119910119894+ 119891119868

ℎ

119894119891ℎ= 0 (120)

where 119888 = 119888(x) is a scalar function on119872

Theorem 27 Every twisted product Finsler manifold(119872

1times119891119872

2 119865) with isotropic mean Berwald curvature is a

weakly Berwald manifold

Proof Suppose that 119865 is isotropic mean Berwald twistedproduct Finsler metric Then differentiating (120) withrespect to V120574 gives us

119888 (119899 + 1) 1198912119865minus5V

120574119910119894= 0 (121)

Thus we conclude that 119888 = 0 This implies that 119865 reduces to aweakly Berwald metric

6 Locally Dually Flat Twisted ProductFinsler Manifolds

In [23] Amari and Nagaoka introduced the notion of duallyflat Riemannian metrics when they study the informationgeometry on Riemannian manifolds Information geometryhas emerged from investigating the geometrical structure ofa family of probability distributions and has been appliedsuccessfully to various areas including statistical inferencecontrol system theory andmultiterminal information theoryIn Finsler geometry Shen extends the notion of locally duallyflatness for Finsler metrics [17] Dually flat Finsler metricsform a special and valuable class of Finsler metrics in Finslerinformation geometry which play a very important role instudying flat Finsler information structure [24 25]

In this section we study locally dually flat twisted productFinsler metrics It is remarkable that a Finsler metric 119865 =119865(x y) on a manifold119872 is said to be locally dually flat if atany point there is a standard coordinate system (x119886 y119886) in119879119872such that it satisfies

12059721198652

120597x119887120597y119886y119887 = 2120597119865

2

120597x119886 (122)

In this case the coordinate (x119886) is called an adapted localcoordinate system Byusing (122) we can obtain the followinglemma

ISRN Geometry 11

Lemma 28 Let (1198721times119891119872

2 119865) be a twisted product Finsler

manifold Then 119865 is locally dually flat if and only if 1198651and 119865

2

satisfy in the following equations

12059721198652

1

120597119909119896120597119910119897119910119896= 2120597119865

2

1

120597119909119897+ 4119891119891

1198971198652

2 (123)

4119891119896V120573119910119896+ 119891

12059721198652

2

120597119906120572120597V120573V120572 + 4119891

120572V120573V120572 = 2119891

1205971198652

2

120597119906120573+ 4119891

1205731198652

2

(124)

Now let 119865 be a locally dually flat Finsler metric Takingderivative with respect to V120574 from (123) yields 119891

119897= 0 which

means that 119891 is a constant function on1198721 In this case the

relations (123) and (124) reduce to the following

12059721198652

1

120597119909119896120597119910119897119910119896= 2120597119865

2

1

120597119909119897 (125)

11989112059721198652

2

120597119906120572120597V120573V120572 + 4119891

120572V120573V120572 = 2119891

1205971198652

2

120597119906120573+ 4119891

1205731198652

2 (126)

By (125) we deduce that 1198651is locally dually flat

Now we assume that 1198651and 119865

2are locally dually flat

Finsler metrics Then we have

12059721198652

1

120597119909119896120597119910119897119910119896= 2120597119865

2

1

120597119909119897

12059721198652

2

120597119906120572120597V120573V120572 = 2

1205971198652

2

120597119906120573

(127)

By (127) we derive that (123) and (124) are hold if and only ifthe following hold

119891119897= 0 119891

120572V120573V120572 = 119891

1205731198652

2 (128)

Therefore we can conclude the following

Theorem 29 Let (1198721times119891119872

2 119865) be a twisted product Finsler

manifold

(i) If 119865 is locally dually flat then 1198651is locally dually flat 119891

is a function with respect to (119906120572) only and 1198652satisfies

in (126)(ii) If 119865

1and 119865

2are locally dually flat then 119865 is locally

dually flat if and only if 119891 is a function with respect(119906

120572) only and 119865

2satisfies in (128)

ByTheorem 29 we conclude the following

Corollary 30 There is not exist any locally dually flat propertwisted product Finsler manifold

References

[1] J K Beem P E Ehrlich and K L Easley Global LorentzianGeometry vol 202 of Monographs and Textbooks in Pure andApplied Mathematics Marcel Dekker New York NY USA 2ndedition 1996

[2] G Ganchev and V Mihova ldquoRiemannian manifolds of quasi-constant sectional curvaturesrdquo Journal fur die Reine und Ange-wandte Mathematik vol 522 pp 119ndash141 2000

[3] B OrsquoNeill Semi-Riemannian Geometry With Applications toRelativity vol 103 of Pure and Applied Mathematics AcademicPress New York NY USA 1983

[4] M Lohnherr and H Reckziegel ldquoOn ruled real hypersurfacesin complex space formsrdquo Geometriae Dedicata vol 74 no 3pp 267ndash286 1999

[5] B-Y Chen F Dillen L Verstraelen and L VranckenldquoLagrangian isometric immersions of a real-space-form119872119899

(119888)

into a complex-space-form119872119899(4119888)rdquoMathematical Proceedings

of the Cambridge Philosophical Society vol 124 no 1 pp 107ndash125 1998

[6] N Koike ldquoThe decomposition of curvature netted hypersur-facesrdquo Geometriae Dedicata vol 54 no 1 pp 1ndash11 1995

[7] B-y Chen Geometry of Submanifolds and its ApplicationsScience University of Tokyo Tokyo Japan 1981

[8] R Ponge and H Reckziegel ldquoTwisted products in pseudo-Riemannian geometryrdquo Geometriae Dedicata vol 48 no 1 pp15ndash25 1993

[9] B-Y Chen ldquoTwisted product CR-submanifolds in Kaehlermanifoldsrdquo Tamsui Oxford Journal of Mathematical Sciencesvol 16 no 2 pp 105ndash121 2000

[10] L Kozma I R Peter and H Shimada ldquoOn the twisted productof Finsler manifoldsrdquo Reports on Mathematical Physics vol 57no 3 pp 375ndash383 2006

[11] M Matsumoto and C Shibata ldquoOn semi-C-reducibility T-tensor= 0 and S4-likeness of Finsler spacesrdquo Journal of Math-ematics of Kyoto University vol 19 no 2 pp 301ndash314 1979

[12] B Najafi Z Shen and A Tayebi ldquoFinsler metrics of scalar flagcurvature with special non-Riemannian curvature propertiesrdquoGeometriae Dedicata vol 131 pp 87ndash97 2008

[13] E Peyghan andAHeydari ldquoConformal vector fields on tangentbundle of a Riemannian manifoldrdquo Journal of MathematicalAnalysis and Applications vol 347 no 1 pp 136ndash142 2008

[14] Z Shen Differential Geometry of Spray and Finsler SpacesKluwer Academic Publishers Dordrecht The Netherlands2001

[15] M Matsumoto and S-I Hojo ldquoA conclusive theorem on C-reducible Finsler spacesrdquo The Tensor Society vol 32 no 2 pp225ndash230 1978

[16] D Bao S-S Chern and Z Shen An Introduction to Riemann-Finsler Geometry vol 200 of Graduate Texts in MathematicsSpringer New York NY USA 2000

[17] Z Shen ldquoRiemann-Finsler geometry with applications to infor-mation geometryrdquo Chinese Annals of Mathematics B vol 27 no1 pp 73ndash94 2006

[18] Y Ichijyo ldquoFinsler manifolds modeled on a Minkowski spacerdquoJournal of Mathematics of Kyoto University vol 16 no 3 pp639ndash652 1976

[19] X Chen and Z Shen ldquoOn Douglas metricsrdquo PublicationesMathematicae Debrecen vol 66 no 3-4 pp 503ndash512 2005

[20] L Berwald ldquoUber Parallelubertragung in Raumen mit allge-meiner Massbestimmungrdquo Jahresbericht der DMVmdashDeutscheMathematiker-Vereinigung vol 34 pp 213ndash220 1926

[21] M Matsumoto ldquoOn Finsler spaces with Randersrsquo metric andspecial forms of important tensorsrdquo Journal of Mathematics ofKyoto University vol 14 pp 477ndash498 1974

12 ISRN Geometry

[22] A Tayebi and E Peyghan ldquoOn special Berwald metricsrdquoSymmetry Integrability and Geometry vol 6 article 008 9pages 2010

[23] S-I Amari and H NagaokaMethods of Information Geometryvol 191 of Translations of Mathematical Monographs AmericanMathematical Society Providence RI USA 2000

[24] X Cheng Z Shen and Y Zhou ldquoOn locally dually flat Finslermetricsrdquo International Journal ofMathematics vol 21 no 11 pp1531ndash1543 2010

[25] Q Xia ldquoOn a class of locally dually flat Finsler metrics ofisotropic flag curvaturerdquo Publicationes Mathematicae Debrecenvol 78 no 1 pp 169ndash190 2011

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

10 ISRN Geometry

Now we are going to study twisted product Finslermanifold of isotropicmean Berwald curvature For this workwe must compute the coefficients of mean Berwald curvatureof a twisted product Finsler manifold

Lemma 23 Let (1198721times119891119872

2 119865) be a twisted product Finsler

manifold Then the coefficients of mean Berwald curvature areas follows

E120572120573= 119864

120572120573+ 119891119892

120572120573119868ℎ119891ℎ+1

2119891119868

]120572120573119891]119865

2

2+ 119891

minus1119891]

times (119862120574]120572120573

V120574+ 119868

]120572V120573+ 119868

]120573V120572+ 119862

]120572120573+ 119868

]119892120572120573)

(112)

E119894119895= 119864

119894119895+1

2119891119868

ℎ

119895119894119891ℎ1198652

2 (113)

E119894120573= 119891119868

ℎ

119894119891ℎV120573 (114)

where 119864119894119895

and 119864120572120573

are the coefficients of mean Berwaldcurvature of (119872

1 119865

1) and (119872

2 119865

2) respectively

Proof By definition and Lemma 19 we get the proof

Theorem24 The twisted product Finslermanifold (1198721times119891119872

2

119865) is weakly Berwald if and only if (1198721 119865

1) is weakly Berwald

119868ℎ119891ℎ= 0 and the following hold

119864120572120573= minus

1

2119891119868

]120572120573119891]119865

2

2minus 119891

minus1119891]

times (119862120574]120572120573

V120574+ 119868

]120572V120573+ 119868

]120573V120572+ 119862

]120572120573+ 119868

]119892120572120573)

(115)

Proof If (1198721times119891119872

2) be a weakly Berwald manifold then we

have

E120572120573= E

119894119895= E

119894120573= 0 (116)

Thus by using (114) we result that 119868ℎ119894119891ℎ= 0 This equation

implies that

119868ℎ

119895119894119891ℎ= 0 119868

ℎ119891ℎ= 0 (117)

By setting these equations in (112) and (113) we conclude that119864119894119895= 0 and 119864

120572120573satisfies in (115)

Now if 119891 is constant on1198722 then (115) implies that 119864

120572120573=

0 Thus we conclude the following

Corollary 25 Let (1198721times119891119872

2 119865) be a twisted product Finsler

manifold and let 119891 be a function on 1198721 only Then

(1198721times119891119872

2 119865) is weakly Berwald if and only if (119872

1 119865

1) and

(1198722 119865

2) are weakly Berwald manifolds and 119868ℎ119891

ℎ= 0

Now we consider twisted product Finsler manifolds withisotropic mean Berwald curvature It is remarkable that as aconsequence of Lemma 23 we have the following

Lemma26 Twisted product Finslermanifold (1198721times119891119872

2 119865) is

isotropic mean Berwald manifold if and only if

119864120572120573+ 119891119892

120572120573119868ℎ119891ℎ+1

2119891119868

]120572120573119891]119865

2

2+ 119891

minus1119891]

times (119862120574]120572120573

V120574+ 119868

]120572V120573+ 119868

]120573V120572+ 119862

]120572120573+ 119868

]119892120572120573)

minus119899 + 1

2119888119891

2119865minus1(119892

120572120573minus1198912

1198652V120572V120573) = 0

(118)

119864119894119895+1

2119891119868

ℎ

119895119894119891ℎ1198652

2minus119899 + 1

2119888119865

minus1(119892

119894119895minus1

1198652119910119894119910119895) = 0 (119)

119888 (119899 + 1) 119865minus3119910119894+ 119891119868

ℎ

119894119891ℎ= 0 (120)

where 119888 = 119888(x) is a scalar function on119872

Theorem 27 Every twisted product Finsler manifold(119872

1times119891119872

2 119865) with isotropic mean Berwald curvature is a

weakly Berwald manifold

Proof Suppose that 119865 is isotropic mean Berwald twistedproduct Finsler metric Then differentiating (120) withrespect to V120574 gives us

119888 (119899 + 1) 1198912119865minus5V

120574119910119894= 0 (121)

Thus we conclude that 119888 = 0 This implies that 119865 reduces to aweakly Berwald metric

6 Locally Dually Flat Twisted ProductFinsler Manifolds

In [23] Amari and Nagaoka introduced the notion of duallyflat Riemannian metrics when they study the informationgeometry on Riemannian manifolds Information geometryhas emerged from investigating the geometrical structure ofa family of probability distributions and has been appliedsuccessfully to various areas including statistical inferencecontrol system theory andmultiterminal information theoryIn Finsler geometry Shen extends the notion of locally duallyflatness for Finsler metrics [17] Dually flat Finsler metricsform a special and valuable class of Finsler metrics in Finslerinformation geometry which play a very important role instudying flat Finsler information structure [24 25]

In this section we study locally dually flat twisted productFinsler metrics It is remarkable that a Finsler metric 119865 =119865(x y) on a manifold119872 is said to be locally dually flat if atany point there is a standard coordinate system (x119886 y119886) in119879119872such that it satisfies

12059721198652

120597x119887120597y119886y119887 = 2120597119865

2

120597x119886 (122)

In this case the coordinate (x119886) is called an adapted localcoordinate system Byusing (122) we can obtain the followinglemma

ISRN Geometry 11

Lemma 28 Let (1198721times119891119872

2 119865) be a twisted product Finsler

manifold Then 119865 is locally dually flat if and only if 1198651and 119865

2

satisfy in the following equations

12059721198652

1

120597119909119896120597119910119897119910119896= 2120597119865

2

1

120597119909119897+ 4119891119891

1198971198652

2 (123)

4119891119896V120573119910119896+ 119891

12059721198652

2

120597119906120572120597V120573V120572 + 4119891

120572V120573V120572 = 2119891

1205971198652

2

120597119906120573+ 4119891

1205731198652

2

(124)

Now let 119865 be a locally dually flat Finsler metric Takingderivative with respect to V120574 from (123) yields 119891

119897= 0 which

means that 119891 is a constant function on1198721 In this case the

relations (123) and (124) reduce to the following

12059721198652

1

120597119909119896120597119910119897119910119896= 2120597119865

2

1

120597119909119897 (125)

11989112059721198652

2

120597119906120572120597V120573V120572 + 4119891

120572V120573V120572 = 2119891

1205971198652

2

120597119906120573+ 4119891

1205731198652

2 (126)

By (125) we deduce that 1198651is locally dually flat

Now we assume that 1198651and 119865

2are locally dually flat

Finsler metrics Then we have

12059721198652

1

120597119909119896120597119910119897119910119896= 2120597119865

2

1

120597119909119897

12059721198652

2

120597119906120572120597V120573V120572 = 2

1205971198652

2

120597119906120573

(127)

By (127) we derive that (123) and (124) are hold if and only ifthe following hold

119891119897= 0 119891

120572V120573V120572 = 119891

1205731198652

2 (128)

Therefore we can conclude the following

Theorem 29 Let (1198721times119891119872

2 119865) be a twisted product Finsler

manifold

(i) If 119865 is locally dually flat then 1198651is locally dually flat 119891

is a function with respect to (119906120572) only and 1198652satisfies

in (126)(ii) If 119865

1and 119865

2are locally dually flat then 119865 is locally

dually flat if and only if 119891 is a function with respect(119906

120572) only and 119865

2satisfies in (128)

ByTheorem 29 we conclude the following

Corollary 30 There is not exist any locally dually flat propertwisted product Finsler manifold

References

[1] J K Beem P E Ehrlich and K L Easley Global LorentzianGeometry vol 202 of Monographs and Textbooks in Pure andApplied Mathematics Marcel Dekker New York NY USA 2ndedition 1996

[2] G Ganchev and V Mihova ldquoRiemannian manifolds of quasi-constant sectional curvaturesrdquo Journal fur die Reine und Ange-wandte Mathematik vol 522 pp 119ndash141 2000

[3] B OrsquoNeill Semi-Riemannian Geometry With Applications toRelativity vol 103 of Pure and Applied Mathematics AcademicPress New York NY USA 1983

[4] M Lohnherr and H Reckziegel ldquoOn ruled real hypersurfacesin complex space formsrdquo Geometriae Dedicata vol 74 no 3pp 267ndash286 1999

[5] B-Y Chen F Dillen L Verstraelen and L VranckenldquoLagrangian isometric immersions of a real-space-form119872119899

(119888)

into a complex-space-form119872119899(4119888)rdquoMathematical Proceedings

of the Cambridge Philosophical Society vol 124 no 1 pp 107ndash125 1998

[6] N Koike ldquoThe decomposition of curvature netted hypersur-facesrdquo Geometriae Dedicata vol 54 no 1 pp 1ndash11 1995

[7] B-y Chen Geometry of Submanifolds and its ApplicationsScience University of Tokyo Tokyo Japan 1981

[8] R Ponge and H Reckziegel ldquoTwisted products in pseudo-Riemannian geometryrdquo Geometriae Dedicata vol 48 no 1 pp15ndash25 1993

[9] B-Y Chen ldquoTwisted product CR-submanifolds in Kaehlermanifoldsrdquo Tamsui Oxford Journal of Mathematical Sciencesvol 16 no 2 pp 105ndash121 2000

[10] L Kozma I R Peter and H Shimada ldquoOn the twisted productof Finsler manifoldsrdquo Reports on Mathematical Physics vol 57no 3 pp 375ndash383 2006

[11] M Matsumoto and C Shibata ldquoOn semi-C-reducibility T-tensor= 0 and S4-likeness of Finsler spacesrdquo Journal of Math-ematics of Kyoto University vol 19 no 2 pp 301ndash314 1979

[12] B Najafi Z Shen and A Tayebi ldquoFinsler metrics of scalar flagcurvature with special non-Riemannian curvature propertiesrdquoGeometriae Dedicata vol 131 pp 87ndash97 2008

[13] E Peyghan andAHeydari ldquoConformal vector fields on tangentbundle of a Riemannian manifoldrdquo Journal of MathematicalAnalysis and Applications vol 347 no 1 pp 136ndash142 2008

[14] Z Shen Differential Geometry of Spray and Finsler SpacesKluwer Academic Publishers Dordrecht The Netherlands2001

[15] M Matsumoto and S-I Hojo ldquoA conclusive theorem on C-reducible Finsler spacesrdquo The Tensor Society vol 32 no 2 pp225ndash230 1978

[16] D Bao S-S Chern and Z Shen An Introduction to Riemann-Finsler Geometry vol 200 of Graduate Texts in MathematicsSpringer New York NY USA 2000

[17] Z Shen ldquoRiemann-Finsler geometry with applications to infor-mation geometryrdquo Chinese Annals of Mathematics B vol 27 no1 pp 73ndash94 2006

[18] Y Ichijyo ldquoFinsler manifolds modeled on a Minkowski spacerdquoJournal of Mathematics of Kyoto University vol 16 no 3 pp639ndash652 1976

[19] X Chen and Z Shen ldquoOn Douglas metricsrdquo PublicationesMathematicae Debrecen vol 66 no 3-4 pp 503ndash512 2005

[20] L Berwald ldquoUber Parallelubertragung in Raumen mit allge-meiner Massbestimmungrdquo Jahresbericht der DMVmdashDeutscheMathematiker-Vereinigung vol 34 pp 213ndash220 1926

[21] M Matsumoto ldquoOn Finsler spaces with Randersrsquo metric andspecial forms of important tensorsrdquo Journal of Mathematics ofKyoto University vol 14 pp 477ndash498 1974

12 ISRN Geometry

[22] A Tayebi and E Peyghan ldquoOn special Berwald metricsrdquoSymmetry Integrability and Geometry vol 6 article 008 9pages 2010

[23] S-I Amari and H NagaokaMethods of Information Geometryvol 191 of Translations of Mathematical Monographs AmericanMathematical Society Providence RI USA 2000

[24] X Cheng Z Shen and Y Zhou ldquoOn locally dually flat Finslermetricsrdquo International Journal ofMathematics vol 21 no 11 pp1531ndash1543 2010

[25] Q Xia ldquoOn a class of locally dually flat Finsler metrics ofisotropic flag curvaturerdquo Publicationes Mathematicae Debrecenvol 78 no 1 pp 169ndash190 2011

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

ISRN Geometry 11

Lemma 28 Let (1198721times119891119872

2 119865) be a twisted product Finsler

manifold Then 119865 is locally dually flat if and only if 1198651and 119865

2

satisfy in the following equations

12059721198652

1

120597119909119896120597119910119897119910119896= 2120597119865

2

1

120597119909119897+ 4119891119891

1198971198652

2 (123)

4119891119896V120573119910119896+ 119891

12059721198652

2

120597119906120572120597V120573V120572 + 4119891

120572V120573V120572 = 2119891

1205971198652

2

120597119906120573+ 4119891

1205731198652

2

(124)

Now let 119865 be a locally dually flat Finsler metric Takingderivative with respect to V120574 from (123) yields 119891

119897= 0 which

means that 119891 is a constant function on1198721 In this case the

relations (123) and (124) reduce to the following

12059721198652

1

120597119909119896120597119910119897119910119896= 2120597119865

2

1

120597119909119897 (125)

11989112059721198652

2

120597119906120572120597V120573V120572 + 4119891

120572V120573V120572 = 2119891

1205971198652

2

120597119906120573+ 4119891

1205731198652

2 (126)

By (125) we deduce that 1198651is locally dually flat

Now we assume that 1198651and 119865

2are locally dually flat

Finsler metrics Then we have

12059721198652

1

120597119909119896120597119910119897119910119896= 2120597119865

2

1

120597119909119897

12059721198652

2

120597119906120572120597V120573V120572 = 2

1205971198652

2

120597119906120573

(127)

By (127) we derive that (123) and (124) are hold if and only ifthe following hold

119891119897= 0 119891

120572V120573V120572 = 119891

1205731198652

2 (128)

Therefore we can conclude the following

Theorem 29 Let (1198721times119891119872

2 119865) be a twisted product Finsler

manifold

(i) If 119865 is locally dually flat then 1198651is locally dually flat 119891

is a function with respect to (119906120572) only and 1198652satisfies

in (126)(ii) If 119865

1and 119865

2are locally dually flat then 119865 is locally

dually flat if and only if 119891 is a function with respect(119906

120572) only and 119865

2satisfies in (128)

ByTheorem 29 we conclude the following

Corollary 30 There is not exist any locally dually flat propertwisted product Finsler manifold

References

[1] J K Beem P E Ehrlich and K L Easley Global LorentzianGeometry vol 202 of Monographs and Textbooks in Pure andApplied Mathematics Marcel Dekker New York NY USA 2ndedition 1996

[2] G Ganchev and V Mihova ldquoRiemannian manifolds of quasi-constant sectional curvaturesrdquo Journal fur die Reine und Ange-wandte Mathematik vol 522 pp 119ndash141 2000

[3] B OrsquoNeill Semi-Riemannian Geometry With Applications toRelativity vol 103 of Pure and Applied Mathematics AcademicPress New York NY USA 1983

[4] M Lohnherr and H Reckziegel ldquoOn ruled real hypersurfacesin complex space formsrdquo Geometriae Dedicata vol 74 no 3pp 267ndash286 1999

[5] B-Y Chen F Dillen L Verstraelen and L VranckenldquoLagrangian isometric immersions of a real-space-form119872119899

(119888)

into a complex-space-form119872119899(4119888)rdquoMathematical Proceedings

of the Cambridge Philosophical Society vol 124 no 1 pp 107ndash125 1998

[6] N Koike ldquoThe decomposition of curvature netted hypersur-facesrdquo Geometriae Dedicata vol 54 no 1 pp 1ndash11 1995

[7] B-y Chen Geometry of Submanifolds and its ApplicationsScience University of Tokyo Tokyo Japan 1981

[8] R Ponge and H Reckziegel ldquoTwisted products in pseudo-Riemannian geometryrdquo Geometriae Dedicata vol 48 no 1 pp15ndash25 1993

[9] B-Y Chen ldquoTwisted product CR-submanifolds in Kaehlermanifoldsrdquo Tamsui Oxford Journal of Mathematical Sciencesvol 16 no 2 pp 105ndash121 2000

[10] L Kozma I R Peter and H Shimada ldquoOn the twisted productof Finsler manifoldsrdquo Reports on Mathematical Physics vol 57no 3 pp 375ndash383 2006

[11] M Matsumoto and C Shibata ldquoOn semi-C-reducibility T-tensor= 0 and S4-likeness of Finsler spacesrdquo Journal of Math-ematics of Kyoto University vol 19 no 2 pp 301ndash314 1979

[12] B Najafi Z Shen and A Tayebi ldquoFinsler metrics of scalar flagcurvature with special non-Riemannian curvature propertiesrdquoGeometriae Dedicata vol 131 pp 87ndash97 2008

[13] E Peyghan andAHeydari ldquoConformal vector fields on tangentbundle of a Riemannian manifoldrdquo Journal of MathematicalAnalysis and Applications vol 347 no 1 pp 136ndash142 2008

[14] Z Shen Differential Geometry of Spray and Finsler SpacesKluwer Academic Publishers Dordrecht The Netherlands2001

[15] M Matsumoto and S-I Hojo ldquoA conclusive theorem on C-reducible Finsler spacesrdquo The Tensor Society vol 32 no 2 pp225ndash230 1978

[16] D Bao S-S Chern and Z Shen An Introduction to Riemann-Finsler Geometry vol 200 of Graduate Texts in MathematicsSpringer New York NY USA 2000

[17] Z Shen ldquoRiemann-Finsler geometry with applications to infor-mation geometryrdquo Chinese Annals of Mathematics B vol 27 no1 pp 73ndash94 2006

[18] Y Ichijyo ldquoFinsler manifolds modeled on a Minkowski spacerdquoJournal of Mathematics of Kyoto University vol 16 no 3 pp639ndash652 1976

[19] X Chen and Z Shen ldquoOn Douglas metricsrdquo PublicationesMathematicae Debrecen vol 66 no 3-4 pp 503ndash512 2005

[20] L Berwald ldquoUber Parallelubertragung in Raumen mit allge-meiner Massbestimmungrdquo Jahresbericht der DMVmdashDeutscheMathematiker-Vereinigung vol 34 pp 213ndash220 1926

[21] M Matsumoto ldquoOn Finsler spaces with Randersrsquo metric andspecial forms of important tensorsrdquo Journal of Mathematics ofKyoto University vol 14 pp 477ndash498 1974

12 ISRN Geometry

[22] A Tayebi and E Peyghan ldquoOn special Berwald metricsrdquoSymmetry Integrability and Geometry vol 6 article 008 9pages 2010

[23] S-I Amari and H NagaokaMethods of Information Geometryvol 191 of Translations of Mathematical Monographs AmericanMathematical Society Providence RI USA 2000

[24] X Cheng Z Shen and Y Zhou ldquoOn locally dually flat Finslermetricsrdquo International Journal ofMathematics vol 21 no 11 pp1531ndash1543 2010

[25] Q Xia ldquoOn a class of locally dually flat Finsler metrics ofisotropic flag curvaturerdquo Publicationes Mathematicae Debrecenvol 78 no 1 pp 169ndash190 2011

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

12 ISRN Geometry

[22] A Tayebi and E Peyghan ldquoOn special Berwald metricsrdquoSymmetry Integrability and Geometry vol 6 article 008 9pages 2010

[23] S-I Amari and H NagaokaMethods of Information Geometryvol 191 of Translations of Mathematical Monographs AmericanMathematical Society Providence RI USA 2000

[24] X Cheng Z Shen and Y Zhou ldquoOn locally dually flat Finslermetricsrdquo International Journal ofMathematics vol 21 no 11 pp1531ndash1543 2010

[25] Q Xia ldquoOn a class of locally dually flat Finsler metrics ofisotropic flag curvaturerdquo Publicationes Mathematicae Debrecenvol 78 no 1 pp 169ndash190 2011

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of