Hindawi Publishing CorporationISRN GeometryVolume 2013 Article ID 932564 6 pageshttpdxdoiorg1011552013932564

Research ArticleOn the M-Projective Curvature Tensor of119873(120581)-ContactMetric Manifolds

R N Singh and Shravan K Pandey

Department of Mathematical Sciences APS University Rewa Madhya Pradesh 486003 India

Correspondence should be addressed to Shravan K Pandey shravanmathgmailcom

Received 29 December 2012 Accepted 20 January 2013

Academic Editors J Keesling A Morozov and E Previato

Copyright copy 2013 R N Singh and S K PandeyThis is an open access article distributed under the Creative Commons AttributionLicense which permits unrestricted use distribution and reproduction in anymedium provided the originalwork is properly cited

The object of the present paper is to study some curvature conditions on 119873(120581)-contact metric manifolds

1 Introduction

The notion of the odd dimensional manifolds with contactand almost contact structures was initiated by Boothby andWong in 1958 rather from topological point of view Sasakiand Hatakeyama reinvestigated them using tensor calculusin 1961 Tanno [1] classified the connected almost contactmetric manifolds whose automorphism groups possess themaximum dimension For such a manifold the sectionalcurvature of plain sections containing 120585 is a constant say 119888He showed that they can be divided into three classes (i)homogeneous normal contact Riemannian manifolds with119888 gt 0 (ii) global Riemannian products of line or a circle with aKahlermanifold of constant holomorphic sectional curvatureif 119888 = 0 and (iii) a warped product space R times119891 C

119899 if 119888 lt 0 Itis known that the manifolds of class (i) are characterized byadmitting a Sasakian structure Kenmotsu [2] characterizedthe differential geometric properties of the manifolds of class(iii) so the structure obtained is now known as Kenmotsustructure In general these structures are not Sasakian [2]

On the other hand in Pokhariyal and Mishra [3] defineda tensor field 119882

lowast on a Riemannian manifold as

1015840119882lowast(119883 119884 119885 119880)

=1015840119877 (119883 119884 119885 119880) minus

1

2 (119899 minus 1)

times [119878 (119884 119885) 119892 (119883119880)

minus 119878 (119883 119885) 119892 (119884 119880) + 119892 (119884 119885) 119878 (119883119880)

minus119892 (119883 119885) 119878 (119884 119880)]

(1)

where 1015840119882lowast(119883 119884 119885 119880) = 119892(119882

lowast(119883 119884)119885119880) and 1015840119877(119883 119884

119885 119880) = 119892(119877(119883 119884)119885119880) Such a tensor field 119882lowast is known

as m-projective curvature tensor Later Ojha [4] definedand studied the properties of m-projective curvature tensorin Sasakian and Khler manifolds He also showed that itbridges the gap between the conformal curvature tensorconharmonic curvature tensor and concircular curvaturetensor on one side and H-projective curvature tensor onthe other Recently m-projective curvature tensor has beenstudied by Chaubey and Ojha [5] Singh et al [6] Singh[7] and many others Motivated by the above studies inthe present paper we study flatness and symmetry propertyof 119873(120581)-contact metric manifolds regarding m-projectivecurvature tensor The present paper is organized as follows

In this paper we study the m-projective curvature tensorof119873(120581)-contactmetricmanifolds In Section 2 someprelimi-nary results are recalled In Section 3 we studym-projectivelysemisymmetric 119873(120581)-contact metric manifolds Section 4deals with m-projectively flat 119873(120581)-contact metric mani-folds 120585-m-projectively flat 119873(120581)-contact metric manifoldsare studied in Section 5 and obtained necessary and sufficientcondition for an 119873(120581)-contact metric manifold to be 120585-m-projectively flat In Section 6 m-projectively recurrent119873(120581)-contact metric manifolds are studied Section 7 is devotedto the study of 119873(120581)-contact metric manifolds satisfying

2 ISRN Geometry

119882lowastsdot 119878 = 0 The last section deals with an 119873(120581)-contact

metric manifolds satisfying 119882lowastsdot 119877 = 0

2 Contact Metric Manifolds

An odd dimensional differentiable manifold 1198722119899+1 is said to

admit an almost contact structure if there exist a tensor field120601 of type (1 1) a vector field 120585 and a 1-form 120578 satisfying

1206012= minus119868 + 120578 otimes 120585 (2)

120578 (120585) = 1 (3)

120601120585 = 0 (4)

120578 ∘ 120601 = 0 (5)

An almost contact metric structure is said to be normal if theinduced almost complex structure 119869 on the product manifold1198722119899+1

times R defined by

119869 (119883 119891119889

119889119905) = (120601119883 minus 119891120585 120578 (119883)

119889

119889119905) (6)

is integrable where 119883 is tangent to 1198722119899+1 119905 is coordinate

of R and 119891 is smooth function on 1198722119899+1

times R Let 119892 be acompatible Riemannianmetric with almost contact structure(120601 120585 120578) that is

119892 (120601119883 120601119884) = 119892 (119883 119884) minus 120578 (119883) 120578 (119884) (7)

Then 1198722119899+1 becomes an almost contact metric manifold

equipped with an almost contact metric structure (120601 120585 120578 119892)From (2) and (7) it can be easily seen that

119892 (119883 120601119884) = minus119892 (120601119883 119884) (8)

119892 (119883 120585) = 120578 (119883) (9)

for all vector fields 119883 and 119884 An almost contact metricstructure becomes a contact metric structure if

119892 (119883 120601119884) = 119889120578 (119883 119884) (10)

for all vector fields 119883 and 119884 The 1-form 120578 is then a contactform and 120585 is its characteristic vector fieldWe define a (1 1)-tensor field ℎ by ℎ = (12)pound120585120601 where pound denotes the Lie-differentiation Then ℎ is symmetric and satisfies ℎ120601 = minus120601ℎWe have 119879119903 sdot ℎ = 119879119903 sdot 120601ℎ = 0 and ℎ120585 = 0 Also

nabla119883120585 = minus120601119883 minus 120601ℎ119883 (11)

holds in a contact metric manifoldA normal contactmetricmanifold is a Sasakianmanifold

An almost contact metric manifold is Sasakian if and only if

(nabla119883120601) (119884) = 119892 (119883 119884) 120585 minus 120578 (119884)119883 (12)

for all vector fields 119883 and 119884 where nabla is the Levi-Civitaconnection of the Riemannian metric 119892 A contact metricmanifold 119872

2119899+1 for which 120585 is a killing vector is said to be

a K-contact manifold A Sasakian manifold is K-contact butthe converse needs not be true However a 3-dimensional K-contact manifold is Sasakian [8] It is well known that thetangent sphere bundle of a flat Riemannian manifold admitsa contact metric structure satisfying 119877(119883 119884)120585 = 0 [9] On theother hand on a Sasakian manifold the following holds

119877 (119883 119884) 120585 = 120578 (119884)119883 minus 120578 (119883)119884 (13)

As a generalization of both 119877(119883 119884)120585 = 0 and the Sasakiancase Blair et al [11] considered the (120581 120583)-nullity conditionon a contact metric manifold and gave several reasons forstudying it

The (120581 120583)-nullity distribution119873(120581 120583) ([10 11]) of contactmetric manifold is defined by

119873(120581 120583) 119901 997888rarr 119873119901 (120581 120583)

= 119885 isin 119879119901119872 119877 (119883 119884)119885

= (120581119868 + 120583ℎ) [119892 (119884 119885)119883 minus 119892 (119883 119885) 119884]

(14)

for all 119883119884 isin 119879119872 where (120581 120583) isin R2 A contact metricmanifold 119872

2119899+1 with 120585 isin 119873(120581 120583) is called a (120581 120583)-manifoldIn particular on a (120581 120583)-manifold we have

119877 (119883 119884) 120585 = 120581 [120578 (119884)119883 minus 120578 (119883)119884]

+ 120583 [120578 (119884) ℎ119883 minus 120578 (119883) ℎ119884]

(15)

On a (120581 120583)-manifold 120581 le 1 If 120581 = 1 the structure isSasakian (ℎ = 0 and 120583 is indeterminate) and if 120581 lt 1 the(120581 120583)-nullity condition determines the curvature of 119872

2119899+1

completely [11] In fact for a (120581 120583)-manifold the conditionsof being a Sasakian manifold a K-contact manifold 120581 = 1

and ℎ = 0 are all equivalentIn a (120581 120583)-manifold the following relations hold ([11 12])

ℎ2= (120581 minus 1)

21206012 120581 le 1

(nabla119883120601) (119884) = 119892 (119883 + ℎ119883 119884) 120585 minus 120578 (119884) (119883 + ℎ119883)

119877 (120585 119883) 119884 = 120581 [119892 (119883 119884) 120585 minus 120578 (119884)119883]

+ 120583 [119892 (ℎ119883 119884) 120585 minus 120578 (119884) ℎ119883]

119878 (119883 120585) = 2119899120581120578 (119883)

119878 (119883 119884) = [2 (119899 minus 1) minus 119899120583] 119892 (119883 119884)

+ [2 (119899 minus 1) + 120583] 119892 (ℎ119883 119884)

+ [2 (1 minus 119899) + 119899 (2120581 + 120583)] 120578 (119883) 120578 (119884)

119899 ge 1

119903 = 2119899 (2119899 minus 2 + 120581 minus 119899120583)

119878 (120601119883 120601119884) = 119878 (119883 119884) minus 2119899120581120578 (119883) 120578 (119884)

minus 2 (2119899 minus 2 + 120583) 119892 (ℎ119883 119884)

(16)

ISRN Geometry 3

where 119878 is the Ricci tensor of type (0 2) 119876 is the Riccioperator that is 119892(119876119883 119884) = 119878(119883 119884) and 119903 is the scalarcurvature of the manifold From (11) it follows that

(nabla119883120578) (119884) = 119892 (119883 + ℎ119883 120601119884) (17)

Also in a (120581 120583)-manifold

120578 (119877 (119883 119884)119885) = 120581 [119892 (119884 119885) 120578 (119883) minus 119892 (119883 119885) 120578 (119884)]

+ 120583 [119892 (ℎ119884 119885) 120578 (119883) minus 119892 (ℎ119883 119885) 120578 (119884)]

(18)

holdsThe 120581-nullity distribution119873(120581) of aRiemannianmanifold

1198722119899+1 [13] is defined by

119873(120581) 119901 997888rarr 119873119901 (120581) = 119885 isin 119879119901119872 119877 (119883 119884)119885

= 120581 (119892 (119884 119885)119883 minus 119892 (119883 119885) 119884)

(19)

for all119883119884 isin 119879119872 and 120581 being a constant If the characteristicvector field 120585 isin 119873(120581) then we call a contact metric manifoldan 119873(120581)-contact metric manifold [14] If 120581 = 1 then 119873(120581)-contact metric manifold is Sasakian and if 120581 = 0 then119873(120581)-contact metric manifold is locally isometric to the product119864119899+1

times 119878119899(4) for 119899 gt 1 and flat for 119899 = 1 If 120581 lt 1 the scalar

curvature is 119903 = 2119899(2119899 minus 2 + 120581) If 120583 = 0 then a (120581 120583)-contactmetric manifold reduces to a119873(120581)-contact metric manifoldsIn [9] 119873(120581)-contact metric manifold were studied in somedetail

In 119873(120581)-contact metric manifolds the following relationshold ([15 16])

ℎ2= (120581 minus 1) 120601

2 120581 le 1 (20)

(nabla119883120601) (119884) = 119892 (119883 + ℎ119883 119884) 120585 minus 120578 (119884) (119883 + ℎ119883) (21)

119877 (120585119883) 119884 = 120581 [119892 (119883 119884) 120585 minus 120578 (119884)119883] (22)

119878 (119883 120585) = 2119899120581120578 (119883) (23)

119878 (119883 119884) = 2 (119899 minus 1) [119892 (119883 119884) + 119892 (ℎ119883 119884)]

+ [2 (1 minus 119899) + 2119899120581] 120578 (119883) 120578 (119884) 119899 ge 1

(24)

119903 = 2119899 (2119899 minus 2 + 120581) (25)

119878 (120601119883 120601119884) = 119878 (119883 119884) minus 2119899120581120578 (119883) 120578 (119884)

minus 4 (119899 minus 1) 119892 (ℎ119883 119884)

(26)

(nabla119883120578) (119884) = 119892 (119883 + ℎ119883 120601119884) (27)

119877 (119883 119884) 120585 = 120581 [120578 (119884)119883 minus 120578 (119883)119884] (28)

120578 (119877 (119883 119884)119885) = 120581 [119892 (119884 119885) 120578 (119883) minus 119892 (119883 119885) 120578 (119884)] (29)

For a (2119899 + 1)-dimensional (119899 gt 1) almost contact metricmanifold m-projective curvature tensor 119882lowast is given by [3]

119882lowast(119883 119884)119885 = 119877 (119883 119884)119885 minus

1

2 (119899 minus 1)

times [119878 (119884 119885)119883 minus 119878 (119883 119885) 119884

+119892 (119884 119885)119876119883 minus 119892 (119883 119885)119876119884]

(30)

for arbitrary vector fields 119883 119884 and 119885 where 119878 is the Riccitensor of type (0 2) and 119876 is the Ricci operator that is119892(119876119883 119884) = 119878(119883 119884)

The m-projective curvature tensor 119882lowast for an 119873(120581)-

contact metric manifold is given by

119882lowast(119883 119884) 120585

= minus120581

(119899 minus 1)[120578 (119884)119883 minus 120578 (119883)119884]

minus1

2 (119899 minus 1)[120578 (119884)119876119883 minus 120578 (119883)119876119884]

(31)

120578 (119882lowast(119883 119884) 120585) = 0 (32)

119882lowast(120585 119884) 119885

= minus119882lowast(119884 120585) 119885

= minus120581

(119899 minus 1)[119892 (119884 119885) 120585 minus 120578 (119885) 119884]

minus1

2 (119899 minus 1)[119878 (119884 119885) 120585 minus 120578 (119885)119876119884]

(33)

120578 (119882lowast(120585 119884) 119885)

= minus120578 (119882lowast(119884 120585) 119885)

= minus120581

(119899 minus 1)[119892 (119884 119885) minus 120578 (119884) 120578 (119885)]

minus1

2 (119899 minus 1)[119878 (119884 119885) minus 2119899120581120578 (119884) 120578 (119885)]

(34)

120578 (119882lowast(119883 119884)119885)

= minus120581

(119899 minus 1)[119892 (119884 119885) 120578 (119883) minus 119892 (119883 119885) 120578 (119884)]

minus1

2 (119899 minus 1)[119878 (119884 119885) 120578 (119883) minus 119878 (119883 119885) 120578 (119884)]

(35)

3 M-Projectively Semisymmetric119873(120581)-Contact Metric Manifolds

Definition 1 A (2119899 + 1)-dimensional (119899 gt 1) 119873(120581)-contactmetric manifold is said to be m-projectively semisymmetric[17] if it satisfies 119877 sdot 119882

lowast= 0 where 119877 is the Riemannian

curvature tensor and119882lowast is them-projective curvature tensor

of the manifold

Theorem 2 An m-projectively semisymmetric 119873(120581)-contactmetric manifold is an Einstein manifold

4 ISRN Geometry

Proof Suppose that an 119873(120581)-contact metric manifold is m-projectively semisymmetric Then we have

(119877 (120585 119883) sdot 119882lowast) (119884 119885)119880 = 0 (36)

The above equation can be written as follows

119877 (120585119883)119882lowast(119884 119885)119880 minus 119882

lowast(119877 (120585 119883) 119884 119885)119880

minus 119882lowast(119884 119877 (120585 119883)119885)119880 minus 119882

lowast(119884 119885) 119877 (120585 119883)119880 = 0

(37)

In view of (22) the above equation reduces to

120581 [119892 (119883119882lowast(119884 119885)119880) 120585 minus 120578 (119882

lowast(119884 119885)119880)119883

minus 119892 (119883 119884)119882lowast(120585 119885)119880 + 120578 (119884)119882

lowast(119883 119885)119880

minus 119892 (119883 119885)119882lowast(119884 120585) 119880 + 120578 (119885)119882

lowast(119884119883)119880

minus119892 (119883119880)119882lowast(119884 119885) 120585 + 120578 (119880)119882

lowast(119884 119885)119883] = 0

(38)

Now taking the inner product of the above equation with 120585

and using (3) and (9) we get

120581 [1015840119882lowast(119884 119885 119880119883) minus 120578 (119882

lowast(119884 119885)119880) 120578 (119883)

minus 119892 (119883 119884) 120578 (119882lowast(120585 119885)119880) + 120578 (119884) 120578 (119882

lowast(119883 119885)119880)

minus 119892 (119883 119885) 120578 (119882lowast(119884 120585) 119880) + 120578 (119885) 120578 (119882

lowast(119884119883)119880)

minus119892 (119883119880) 120578 (119882lowast(119884 119885) 120585) + 120578 (119880) 120578 (119882

lowast(119884 119885)119883) ] = 0

(39)

which on using (30) (32) (34) and (35) gives

120581 [1015840119877 (119884 119885 119880119883) minus

1

2 (119899 minus 1)

times 119878 (119884119883) 119892 (119885119880) minus 119878 (119883 119885) 119892 (119884 119880)

+ 119878 (119883 119885) 120578 (119884) 120578 (119880) minus119878 (119883 119884) 120578 (119885) 120578 (119880)

+120581

(119899 minus 1)

times 119892 (119885119880) 119892 (119883 119884)

minus 119892 (119884119880) 119892 (119883 119885) + 119899119892 (119883 119885) 120578 (119884) 120578 (119880)

minus119899119892 (119883 119884) 120578 (119885) 120578 (119880) ] = 0

(40)

Putting 119885 = 119880 = 119890119894 in the above equation and takingsummation over 119894 1 le 119894 le 2119899 + 1 we get

119878 (119883 119884) = 2119899120581119892 (119883 119884) (41)

which shows that 1198722119899+1 is an Einstein manifold This com-pletes the proof

4 M-Projectively Flat 119873(120581)-ContactMetric Manifolds

Theorem 3 An m-projectively flat 119873(120581)-contact metric man-ifold 119872

2119899+1 is an Einstein manifold

Proof Let 119882lowast(119883 119884 119885 119880) = 0 Then from (30) we have1015840119877 (119883 119884 119885 119880)

=1

2 (119899 minus 1)[119878 (119884 119885) 119892 (119883119880)

minus 119878 (119883 119885) 119892 (119884 119880) + 119892 (119884 119885) 119878 (119883119880)

minus119892 (119883 119885) 119892 (119884 119880)]

(42)

Let 119890119894 be an orthonormal basis of the tangent space at anypoint Putting119884 = 119885 = 119890119894 in the above equation and summingover 119894 1 le 119894 le 2119899 + 1 we get

119878 (119883 119884) = minus119903119892 (119883 119884) (43)

which shows that 1198722119899+1 is an Einstein manifold This com-pletes the proof

5 120585-M-Projectively Flat 119873(120581)-ContactMetric Manifolds

Definition 4 A (2119899 + 1)-dimensional (119899 gt 1) 119873(120581)-contactmetric manifold is said to be 120585-m-projectively flat [18] if119882lowast(119883 119884)120585 = 0 for all 119883119884 isin 119879119872

Theorem 5 A (2119899 + 1)-dimensional (119899 gt 1) 119873(120581)-contactmetric manifold is 120585-m-projectively flat if and only if it is an120578-Einstein manifold

Proof Let 119882lowast(119883 119884)120585 = 0 Then in view if (30) we have

119877 (119883 119884) 120585 =1

2 (119899 minus 1)[119878 (119884 120585)119883 minus 119878 (119883 120585) 119884

+119892 (119884 120585) 119876119883 minus 119892 (119883 120585) 119876119884]

(44)

By virtue of (9) (23) and (28) the above equation reduces to1

2[120578 (119884)119876119883 minus 120578 (119883)119876119884] minus 120581 [120578 (119884)119883 minus 120578 (119883)119884] = 0 (45)

which by putting 119884 = 120585 gives119876119883 = 2120581 [minus119883 + (119899 + 1) 120578 (119883) 120585] (46)

Now taking the inner product of above equation with 119880 weget

119878 (119883119880) = 2120581 [minus119892 (119883119880) + (119899 + 1) 120578 (119883) 120578 (119880)] (47)which shows that 119873(120581)-contact metric manifold is an 120578-Einstein manifold Conversely suppose that (47) is satisfiedThen by virtue of (46) and (31) we have119882

lowast(119883 119884)120585 = 0This

completes the proof

6 M-Projectively Recurrent 119873(120581)-ContactMetric Manifolds

Definition 6 A nonflat Riemannian manifold 1198722119899+1 is said

to be m-projectively recurrent if its m-projective curvaturetensor 119882lowast satisfies the condition

nabla119882lowast= 119860 otimes 119882

lowast (48)

where 119860 is nonzero 1-form

ISRN Geometry 5

Theorem 7 If an 119873(120581)-contact metric manifold is m-projectively recurrent then it is an 120578-Einstein manifold

Proof We define a function 1198912

= 119892(119882lowast119882lowast) on 119872

2119899+1where the metric 119892 is extended to the inner product betweenthe tensor fields Then we have

119891 (119884119891) = 1198912119860 (119884) (49)

This can be written as

119884119891 = 119891 (119860 (119884)) (119891 = 0) (50)

From the above equation we have

119883(119884119891) minus 119884 (119883119891) = 119883119860 (119884) minus 119884119860 (119883) minus 119860 ([119883 119884]) 119891 (51)

Since the left-hand side of the above equation is identicallyzero and 119891 = 0 on 119872

2119899+1 then

119889119860 (119883 119884) = 0 (52)

that is 1-form 119860 is closedNow from

(nabla119884119882lowast) (119885 119880)119881 = 119860 (119884)119882

lowast(119885 119880)119881 (53)

we have

(nabla119883nabla119884119882lowast) (119885 119880)119881 = 119883119860 (119884) + 119860 (119883)119860 (119884)119882

lowast(119885 119880)119881

(54)

In view of (52) and (54) we have

(119877 (119883 119884) sdot 119882lowast) (119885 119880)119881 = [2119889119860 (119883 119884)]119882

lowast(119885 119880)119881

= 0

(55)

Thus by virtue of Theorem 3 the above equation shows that1198722119899+1 is an 120578-Einstein manifold This completes the proof

7 119873(120581)-Contact Metric Manifolds Satisfying119882lowastsdot 119878 = 0

Theorem8 If on an119873(120581)-contact metric manifold119882lowastsdot119878 = 0

then 119878(119876119883 119884) = 2120581[(119899 minus 1)119878(119883119884) + 2119899120581119892(119883 119884)]

Proof Let 119882lowast(120585 119883) sdot 119878 = 0 In this case we can write

119878 (119882lowast(120585 119883) 119884 119885) + 119878 (119884119882

lowast(120585 119883)119885) = 0 (56)

In view of (34) the above equation reduces to

minus 120581 [2119899120581 119892 (119883 119884) 120578 (119885) + 119892 (119883 119885) 120578 (119884)

minus 120578 (119884) 119878 (119883 119885) + 120578 (119885) 119878 (119883 119884)]

+1

2[2119899120581 119878 (119883 119884) 120578 (119885) + 119878 (119883 119885) 120578 (119884)

minus 120578 (119884) 119878 (119876119883119885) + 120578 (119885) 119878 (119876119883 119884)] = 0

(57)

Now putting 119885 = 120585 in above equation and using (3) (9) and(23) we get

119878 (119876119883 119884) = 2120581 [(119899 minus 1) 119878 (119883 119884) + 2119899120581119892 (119883 119884)] (58)

This completes the proof

8 119873(120581)-Contact Metric Manifolds Satisfying119882lowastsdot 119877 = 0

Theorem 9 On an119873(120581)-contact metric manifold if119882lowast sdot 119877 =

0 then 119878(119876119883 119884) = 2120581[(119899 minus 1)119878(119883 119884) + 2119899120581119892(119883 119884)]

Proof Suppose that 119882lowast(120585 119883) sdot 119877 = 0 then it can be writtenas

119882lowast(120585 119883) 119877 (119884 119885)119880 minus 119877 (119882

lowast(120585 119883) 119884 119885)119880

minus 119877 (119884119882lowast(120585 119883)119885)119880 minus 119877 (119884 119885)119882

lowast(120585 119883)119880 = 0

(59)

which on using (33) takes the form

minus120581

(119899 minus 1)[119892 (119883 119877 (119884 119885)119880) 120585 minus 120578 (119877 (119884 119885)119880)119883

minus 119892 (119883 119884) 119877 (120585 119885)119880 + 120578 (119884) 119877 (119883 119885)119880

minus 119892 (119883 119885) 119877 (119884 120585) 119880 + 120578 (119885) 119877 (119884119883)119880

minus119892 (119883119880) 119877 (119884 119885) 120585 + 120578 (119880) 119877 (119884 119885)119883]

minus1

2 (119899 minus 1)[1015840119877 (119884 119885119880 119876119883) 120585 minus 120578 (119877 (119884 119885)119880)119876119883

minus 119878 (119883 119884) 119877 (120585 119885)119880 + 120578 (119884) 119877 (119876119883119885)119880

minus 119878 (119883 119885) 119877 (119884 120585)119880 + 120578 (119885) 119877 (119884 119876119883)119880

minus119878 (119883119880) 119877 (119884 119885) 120585 + 120578 (119880) 119877 (119884 119885)119876119883]

= 0

(60)

Taking the inner product of above equation with 120585 we get

minus120581

(119899 minus 1)[1015840119877 (119884 119885 119880119883) minus 119892 (119883 119884) 120578 (119877 (120585 119885)119880)

+ 120578 (119884) 120578 (119877 (119883 119885)119880) minus 119892 (119883 119885) 120578 (119877 (119884 120585) 119880)

+ 120578 (119885) 120578 (119877 (119884119883)119880) minus 119892 (119883119880) 120578 (119877 (119884 119885) 120585)

+120578 (119880) 120578 (119877 (119884 119885)119883) ]

minus1

2 (119899 minus 1)[1015840119877 (119884 119885119880119876119883) minus 119878 (119883 119884) 120578 (119877 (120585 119885)119880)

+ 120578 (119884) 120578 (119877 (119876119883119885)119880) minus 119878 (119883 119885) 120578 (119877 (119884 120585)119880)

+ 120578 (119885) 120578 (119877 (119884 119876119883)119880) minus 119878 (119883119880) 120578 (119877 (119884 119885) 120585)

+120578 (119880) 120578 (119877 (119884 119885)119876119883) ] = 0

(61)

Now using (22) (28) and (29) in the above equation we get

minus120581

(119899 minus 1)[1015840119877 (119884 119885119880119883) + 120581 119892 (119883 119885) 119892 (119884 119880)

minus119892 (119883 119884) 119892 (119885119880) ]

6 ISRN Geometry

minus1

2 (119899 minus 1)[1015840119877 (119884 119885119880 119876119883) + 120581 119878 (119883 119885) 119892 (119884 119880)

minus119878 (119883 119884) 119892 (119885119880) ] = 0

(62)

Putting 119885 = 119880 = 119890119894 in the above equation and summing over119894 1 le 119894 le 2119899 + 1 we get

119878 (119876119883 119884) = 2120581 [(119899 minus 1) 119878 (119883 119884) + 2119899120581119892 (119883 119884)] (63)

This completes the proof

References

[1] S Tanno ldquoThe automorphism groups of almost contact Rie-mannian manifoldsrdquoThe Tohoku Mathematical Journal vol 21pp 21ndash38 1969

[2] K Kenmotsu ldquoA class of almost contact Riemannian mani-foldsrdquo The Tohoku Mathematical Journal vol 24 pp 93ndash1031972

[3] G P Pokhariyal and R S Mishra ldquoCurvature tensorsrsquo and theirrelativistics significancerdquo Yokohama Mathematical Journal vol18 pp 105ndash108 1970

[4] R H Ojha ldquoM-projectively flat Sasakian manifoldsrdquo IndianJournal of Pure and Applied Mathematics vol 17 no 4 pp 481ndash484 1986

[5] S K Chaubey and R H Ojha ldquoOn the m-projective curvaturetensor of a Kenmotsu manifoldrdquo Differential Geometry vol 12pp 52ndash60 2010

[6] R N Singh S K Pandey and G Pandey ldquoOn a type ofKenmotsu manifoldrdquo Bulletin of Mathematical Analysis andApplications vol 4 no 1 pp 117ndash132 2012

[7] J P Singh ldquoOn m-projective recurrent Riemannian manifoldrdquoInternational Journal ofMathematical Analysis vol 6 no 24 pp1173ndash1178 2012

[8] J-B Jun I B Kim and U K Kim ldquoOn 3-dimensional almostcontact metric manifoldsrdquo Kyungpook Mathematical Journalvol 34 no 2 pp 293ndash301 1994

[9] C Baikoussis D E Blair and T Koufogiorgos ldquoA decompo-sition of the curvature tensor of a contact manifold satisfying119877(119883 119884)120585 = 120581[120578(119884)119883minus120578(119883)119884]rdquo Mathematics Technical ReportUniversity of Ioanniana 1992

[10] B J Papantoniou ldquoContact Riemannian manifolds satisfying119877(120585119883)119877 = 0 and 120585 isin (120581 120583)-nullity distributionrdquo YokohamaMathematical Journal vol 40 no 2 pp 149ndash161 1993

[11] D E Blair T Koufogiorgos and B J Papantoniou ldquoContactmetric manifolds satisfying a nullity conditionrdquo Israel Journalof Mathematics vol 91 no 1ndash3 pp 189ndash214 1995

[12] E Boeckx ldquoA full classification of contact metric (120581 120583)-spacesrdquoIllinois Journal of Mathematics vol 44 no 1 pp 212ndash219 2000

[13] S Tanno ldquoRicci curvatures of contact Riemannian manifoldsrdquoThe Tohoku Mathematical Journal vol 40 no 3 pp 441ndash4481988

[14] D E Blair J-S Kim and M M Tripathi ldquoOn the concircularcurvature tensor of a contact metric manifoldrdquo Journal of theKorean Mathematical Society vol 42 no 5 pp 883ndash892 2005

[15] D E Blair T Koufogiorgos and R Sharma ldquoA classification of3-dimensional contact metric manifolds with119876120593 = 120593119876rdquo KodaiMathematical Journal vol 13 no 3 pp 391ndash401 1990

[16] D E Blair and H Chen ldquoA classification of 3-dimensionalcontact metric manifolds with 119876120593 = 120593119876 IIrdquo Bulletin of theInstitute of Mathematics vol 20 no 4 pp 379ndash383 1992

[17] U C De and A Sarkar ldquoOn a type of P-Sasakian manifoldsrdquoMathematical Reports vol 11(61) no 2 pp 139ndash144 2009

[18] G Zhen J L Cabrerizo L M Fernandez and M FernandezldquoOn 120585-conformally flat contact metric manifoldsrdquo Indian Jour-nal of Pure and AppliedMathematics vol 28 no 6 pp 725ndash7341997

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Stochastic AnalysisInternational Journal of

2 ISRN Geometry

119882lowastsdot 119878 = 0 The last section deals with an 119873(120581)-contact

metric manifolds satisfying 119882lowastsdot 119877 = 0

2 Contact Metric Manifolds

An odd dimensional differentiable manifold 1198722119899+1 is said to

admit an almost contact structure if there exist a tensor field120601 of type (1 1) a vector field 120585 and a 1-form 120578 satisfying

1206012= minus119868 + 120578 otimes 120585 (2)

120578 (120585) = 1 (3)

120601120585 = 0 (4)

120578 ∘ 120601 = 0 (5)

An almost contact metric structure is said to be normal if theinduced almost complex structure 119869 on the product manifold1198722119899+1

times R defined by

119869 (119883 119891119889

119889119905) = (120601119883 minus 119891120585 120578 (119883)

119889

119889119905) (6)

is integrable where 119883 is tangent to 1198722119899+1 119905 is coordinate

of R and 119891 is smooth function on 1198722119899+1

times R Let 119892 be acompatible Riemannianmetric with almost contact structure(120601 120585 120578) that is

119892 (120601119883 120601119884) = 119892 (119883 119884) minus 120578 (119883) 120578 (119884) (7)

Then 1198722119899+1 becomes an almost contact metric manifold

equipped with an almost contact metric structure (120601 120585 120578 119892)From (2) and (7) it can be easily seen that

119892 (119883 120601119884) = minus119892 (120601119883 119884) (8)

119892 (119883 120585) = 120578 (119883) (9)

for all vector fields 119883 and 119884 An almost contact metricstructure becomes a contact metric structure if

119892 (119883 120601119884) = 119889120578 (119883 119884) (10)

for all vector fields 119883 and 119884 The 1-form 120578 is then a contactform and 120585 is its characteristic vector fieldWe define a (1 1)-tensor field ℎ by ℎ = (12)pound120585120601 where pound denotes the Lie-differentiation Then ℎ is symmetric and satisfies ℎ120601 = minus120601ℎWe have 119879119903 sdot ℎ = 119879119903 sdot 120601ℎ = 0 and ℎ120585 = 0 Also

nabla119883120585 = minus120601119883 minus 120601ℎ119883 (11)

holds in a contact metric manifoldA normal contactmetricmanifold is a Sasakianmanifold

An almost contact metric manifold is Sasakian if and only if

(nabla119883120601) (119884) = 119892 (119883 119884) 120585 minus 120578 (119884)119883 (12)

for all vector fields 119883 and 119884 where nabla is the Levi-Civitaconnection of the Riemannian metric 119892 A contact metricmanifold 119872

2119899+1 for which 120585 is a killing vector is said to be

a K-contact manifold A Sasakian manifold is K-contact butthe converse needs not be true However a 3-dimensional K-contact manifold is Sasakian [8] It is well known that thetangent sphere bundle of a flat Riemannian manifold admitsa contact metric structure satisfying 119877(119883 119884)120585 = 0 [9] On theother hand on a Sasakian manifold the following holds

119877 (119883 119884) 120585 = 120578 (119884)119883 minus 120578 (119883)119884 (13)

As a generalization of both 119877(119883 119884)120585 = 0 and the Sasakiancase Blair et al [11] considered the (120581 120583)-nullity conditionon a contact metric manifold and gave several reasons forstudying it

The (120581 120583)-nullity distribution119873(120581 120583) ([10 11]) of contactmetric manifold is defined by

119873(120581 120583) 119901 997888rarr 119873119901 (120581 120583)

= 119885 isin 119879119901119872 119877 (119883 119884)119885

= (120581119868 + 120583ℎ) [119892 (119884 119885)119883 minus 119892 (119883 119885) 119884]

(14)

for all 119883119884 isin 119879119872 where (120581 120583) isin R2 A contact metricmanifold 119872

2119899+1 with 120585 isin 119873(120581 120583) is called a (120581 120583)-manifoldIn particular on a (120581 120583)-manifold we have

119877 (119883 119884) 120585 = 120581 [120578 (119884)119883 minus 120578 (119883)119884]

+ 120583 [120578 (119884) ℎ119883 minus 120578 (119883) ℎ119884]

(15)

On a (120581 120583)-manifold 120581 le 1 If 120581 = 1 the structure isSasakian (ℎ = 0 and 120583 is indeterminate) and if 120581 lt 1 the(120581 120583)-nullity condition determines the curvature of 119872

2119899+1

completely [11] In fact for a (120581 120583)-manifold the conditionsof being a Sasakian manifold a K-contact manifold 120581 = 1

and ℎ = 0 are all equivalentIn a (120581 120583)-manifold the following relations hold ([11 12])

ℎ2= (120581 minus 1)

21206012 120581 le 1

(nabla119883120601) (119884) = 119892 (119883 + ℎ119883 119884) 120585 minus 120578 (119884) (119883 + ℎ119883)

119877 (120585 119883) 119884 = 120581 [119892 (119883 119884) 120585 minus 120578 (119884)119883]

+ 120583 [119892 (ℎ119883 119884) 120585 minus 120578 (119884) ℎ119883]

119878 (119883 120585) = 2119899120581120578 (119883)

119878 (119883 119884) = [2 (119899 minus 1) minus 119899120583] 119892 (119883 119884)

+ [2 (119899 minus 1) + 120583] 119892 (ℎ119883 119884)

+ [2 (1 minus 119899) + 119899 (2120581 + 120583)] 120578 (119883) 120578 (119884)

119899 ge 1

119903 = 2119899 (2119899 minus 2 + 120581 minus 119899120583)

119878 (120601119883 120601119884) = 119878 (119883 119884) minus 2119899120581120578 (119883) 120578 (119884)

minus 2 (2119899 minus 2 + 120583) 119892 (ℎ119883 119884)

(16)

ISRN Geometry 3

where 119878 is the Ricci tensor of type (0 2) 119876 is the Riccioperator that is 119892(119876119883 119884) = 119878(119883 119884) and 119903 is the scalarcurvature of the manifold From (11) it follows that

(nabla119883120578) (119884) = 119892 (119883 + ℎ119883 120601119884) (17)

Also in a (120581 120583)-manifold

120578 (119877 (119883 119884)119885) = 120581 [119892 (119884 119885) 120578 (119883) minus 119892 (119883 119885) 120578 (119884)]

+ 120583 [119892 (ℎ119884 119885) 120578 (119883) minus 119892 (ℎ119883 119885) 120578 (119884)]

(18)

holdsThe 120581-nullity distribution119873(120581) of aRiemannianmanifold

1198722119899+1 [13] is defined by

119873(120581) 119901 997888rarr 119873119901 (120581) = 119885 isin 119879119901119872 119877 (119883 119884)119885

= 120581 (119892 (119884 119885)119883 minus 119892 (119883 119885) 119884)

(19)

for all119883119884 isin 119879119872 and 120581 being a constant If the characteristicvector field 120585 isin 119873(120581) then we call a contact metric manifoldan 119873(120581)-contact metric manifold [14] If 120581 = 1 then 119873(120581)-contact metric manifold is Sasakian and if 120581 = 0 then119873(120581)-contact metric manifold is locally isometric to the product119864119899+1

times 119878119899(4) for 119899 gt 1 and flat for 119899 = 1 If 120581 lt 1 the scalar

curvature is 119903 = 2119899(2119899 minus 2 + 120581) If 120583 = 0 then a (120581 120583)-contactmetric manifold reduces to a119873(120581)-contact metric manifoldsIn [9] 119873(120581)-contact metric manifold were studied in somedetail

In 119873(120581)-contact metric manifolds the following relationshold ([15 16])

ℎ2= (120581 minus 1) 120601

2 120581 le 1 (20)

(nabla119883120601) (119884) = 119892 (119883 + ℎ119883 119884) 120585 minus 120578 (119884) (119883 + ℎ119883) (21)

119877 (120585119883) 119884 = 120581 [119892 (119883 119884) 120585 minus 120578 (119884)119883] (22)

119878 (119883 120585) = 2119899120581120578 (119883) (23)

119878 (119883 119884) = 2 (119899 minus 1) [119892 (119883 119884) + 119892 (ℎ119883 119884)]

+ [2 (1 minus 119899) + 2119899120581] 120578 (119883) 120578 (119884) 119899 ge 1

(24)

119903 = 2119899 (2119899 minus 2 + 120581) (25)

119878 (120601119883 120601119884) = 119878 (119883 119884) minus 2119899120581120578 (119883) 120578 (119884)

minus 4 (119899 minus 1) 119892 (ℎ119883 119884)

(26)

(nabla119883120578) (119884) = 119892 (119883 + ℎ119883 120601119884) (27)

119877 (119883 119884) 120585 = 120581 [120578 (119884)119883 minus 120578 (119883)119884] (28)

120578 (119877 (119883 119884)119885) = 120581 [119892 (119884 119885) 120578 (119883) minus 119892 (119883 119885) 120578 (119884)] (29)

For a (2119899 + 1)-dimensional (119899 gt 1) almost contact metricmanifold m-projective curvature tensor 119882lowast is given by [3]

119882lowast(119883 119884)119885 = 119877 (119883 119884)119885 minus

1

2 (119899 minus 1)

times [119878 (119884 119885)119883 minus 119878 (119883 119885) 119884

+119892 (119884 119885)119876119883 minus 119892 (119883 119885)119876119884]

(30)

for arbitrary vector fields 119883 119884 and 119885 where 119878 is the Riccitensor of type (0 2) and 119876 is the Ricci operator that is119892(119876119883 119884) = 119878(119883 119884)

The m-projective curvature tensor 119882lowast for an 119873(120581)-

contact metric manifold is given by

119882lowast(119883 119884) 120585

= minus120581

(119899 minus 1)[120578 (119884)119883 minus 120578 (119883)119884]

minus1

2 (119899 minus 1)[120578 (119884)119876119883 minus 120578 (119883)119876119884]

(31)

120578 (119882lowast(119883 119884) 120585) = 0 (32)

119882lowast(120585 119884) 119885

= minus119882lowast(119884 120585) 119885

= minus120581

(119899 minus 1)[119892 (119884 119885) 120585 minus 120578 (119885) 119884]

minus1

2 (119899 minus 1)[119878 (119884 119885) 120585 minus 120578 (119885)119876119884]

(33)

120578 (119882lowast(120585 119884) 119885)

= minus120578 (119882lowast(119884 120585) 119885)

= minus120581

(119899 minus 1)[119892 (119884 119885) minus 120578 (119884) 120578 (119885)]

minus1

2 (119899 minus 1)[119878 (119884 119885) minus 2119899120581120578 (119884) 120578 (119885)]

(34)

120578 (119882lowast(119883 119884)119885)

= minus120581

(119899 minus 1)[119892 (119884 119885) 120578 (119883) minus 119892 (119883 119885) 120578 (119884)]

minus1

2 (119899 minus 1)[119878 (119884 119885) 120578 (119883) minus 119878 (119883 119885) 120578 (119884)]

(35)

3 M-Projectively Semisymmetric119873(120581)-Contact Metric Manifolds

Definition 1 A (2119899 + 1)-dimensional (119899 gt 1) 119873(120581)-contactmetric manifold is said to be m-projectively semisymmetric[17] if it satisfies 119877 sdot 119882

lowast= 0 where 119877 is the Riemannian

curvature tensor and119882lowast is them-projective curvature tensor

of the manifold

Theorem 2 An m-projectively semisymmetric 119873(120581)-contactmetric manifold is an Einstein manifold

4 ISRN Geometry

Proof Suppose that an 119873(120581)-contact metric manifold is m-projectively semisymmetric Then we have

(119877 (120585 119883) sdot 119882lowast) (119884 119885)119880 = 0 (36)

The above equation can be written as follows

119877 (120585119883)119882lowast(119884 119885)119880 minus 119882

lowast(119877 (120585 119883) 119884 119885)119880

minus 119882lowast(119884 119877 (120585 119883)119885)119880 minus 119882

lowast(119884 119885) 119877 (120585 119883)119880 = 0

(37)

In view of (22) the above equation reduces to

120581 [119892 (119883119882lowast(119884 119885)119880) 120585 minus 120578 (119882

lowast(119884 119885)119880)119883

minus 119892 (119883 119884)119882lowast(120585 119885)119880 + 120578 (119884)119882

lowast(119883 119885)119880

minus 119892 (119883 119885)119882lowast(119884 120585) 119880 + 120578 (119885)119882

lowast(119884119883)119880

minus119892 (119883119880)119882lowast(119884 119885) 120585 + 120578 (119880)119882

lowast(119884 119885)119883] = 0

(38)

Now taking the inner product of the above equation with 120585

and using (3) and (9) we get

120581 [1015840119882lowast(119884 119885 119880119883) minus 120578 (119882

lowast(119884 119885)119880) 120578 (119883)

minus 119892 (119883 119884) 120578 (119882lowast(120585 119885)119880) + 120578 (119884) 120578 (119882

lowast(119883 119885)119880)

minus 119892 (119883 119885) 120578 (119882lowast(119884 120585) 119880) + 120578 (119885) 120578 (119882

lowast(119884119883)119880)

minus119892 (119883119880) 120578 (119882lowast(119884 119885) 120585) + 120578 (119880) 120578 (119882

lowast(119884 119885)119883) ] = 0

(39)

which on using (30) (32) (34) and (35) gives

120581 [1015840119877 (119884 119885 119880119883) minus

1

2 (119899 minus 1)

times 119878 (119884119883) 119892 (119885119880) minus 119878 (119883 119885) 119892 (119884 119880)

+ 119878 (119883 119885) 120578 (119884) 120578 (119880) minus119878 (119883 119884) 120578 (119885) 120578 (119880)

+120581

(119899 minus 1)

times 119892 (119885119880) 119892 (119883 119884)

minus 119892 (119884119880) 119892 (119883 119885) + 119899119892 (119883 119885) 120578 (119884) 120578 (119880)

minus119899119892 (119883 119884) 120578 (119885) 120578 (119880) ] = 0

(40)

Putting 119885 = 119880 = 119890119894 in the above equation and takingsummation over 119894 1 le 119894 le 2119899 + 1 we get

119878 (119883 119884) = 2119899120581119892 (119883 119884) (41)

which shows that 1198722119899+1 is an Einstein manifold This com-pletes the proof

4 M-Projectively Flat 119873(120581)-ContactMetric Manifolds

Theorem 3 An m-projectively flat 119873(120581)-contact metric man-ifold 119872

2119899+1 is an Einstein manifold

Proof Let 119882lowast(119883 119884 119885 119880) = 0 Then from (30) we have1015840119877 (119883 119884 119885 119880)

=1

2 (119899 minus 1)[119878 (119884 119885) 119892 (119883119880)

minus 119878 (119883 119885) 119892 (119884 119880) + 119892 (119884 119885) 119878 (119883119880)

minus119892 (119883 119885) 119892 (119884 119880)]

(42)

Let 119890119894 be an orthonormal basis of the tangent space at anypoint Putting119884 = 119885 = 119890119894 in the above equation and summingover 119894 1 le 119894 le 2119899 + 1 we get

119878 (119883 119884) = minus119903119892 (119883 119884) (43)

which shows that 1198722119899+1 is an Einstein manifold This com-pletes the proof

5 120585-M-Projectively Flat 119873(120581)-ContactMetric Manifolds

Definition 4 A (2119899 + 1)-dimensional (119899 gt 1) 119873(120581)-contactmetric manifold is said to be 120585-m-projectively flat [18] if119882lowast(119883 119884)120585 = 0 for all 119883119884 isin 119879119872

Theorem 5 A (2119899 + 1)-dimensional (119899 gt 1) 119873(120581)-contactmetric manifold is 120585-m-projectively flat if and only if it is an120578-Einstein manifold

Proof Let 119882lowast(119883 119884)120585 = 0 Then in view if (30) we have

119877 (119883 119884) 120585 =1

2 (119899 minus 1)[119878 (119884 120585)119883 minus 119878 (119883 120585) 119884

+119892 (119884 120585) 119876119883 minus 119892 (119883 120585) 119876119884]

(44)

By virtue of (9) (23) and (28) the above equation reduces to1

2[120578 (119884)119876119883 minus 120578 (119883)119876119884] minus 120581 [120578 (119884)119883 minus 120578 (119883)119884] = 0 (45)

which by putting 119884 = 120585 gives119876119883 = 2120581 [minus119883 + (119899 + 1) 120578 (119883) 120585] (46)

Now taking the inner product of above equation with 119880 weget

119878 (119883119880) = 2120581 [minus119892 (119883119880) + (119899 + 1) 120578 (119883) 120578 (119880)] (47)which shows that 119873(120581)-contact metric manifold is an 120578-Einstein manifold Conversely suppose that (47) is satisfiedThen by virtue of (46) and (31) we have119882

lowast(119883 119884)120585 = 0This

completes the proof

6 M-Projectively Recurrent 119873(120581)-ContactMetric Manifolds

Definition 6 A nonflat Riemannian manifold 1198722119899+1 is said

to be m-projectively recurrent if its m-projective curvaturetensor 119882lowast satisfies the condition

nabla119882lowast= 119860 otimes 119882

lowast (48)

where 119860 is nonzero 1-form

ISRN Geometry 5

Theorem 7 If an 119873(120581)-contact metric manifold is m-projectively recurrent then it is an 120578-Einstein manifold

Proof We define a function 1198912

= 119892(119882lowast119882lowast) on 119872

2119899+1where the metric 119892 is extended to the inner product betweenthe tensor fields Then we have

119891 (119884119891) = 1198912119860 (119884) (49)

This can be written as

119884119891 = 119891 (119860 (119884)) (119891 = 0) (50)

From the above equation we have

119883(119884119891) minus 119884 (119883119891) = 119883119860 (119884) minus 119884119860 (119883) minus 119860 ([119883 119884]) 119891 (51)

Since the left-hand side of the above equation is identicallyzero and 119891 = 0 on 119872

2119899+1 then

119889119860 (119883 119884) = 0 (52)

that is 1-form 119860 is closedNow from

(nabla119884119882lowast) (119885 119880)119881 = 119860 (119884)119882

lowast(119885 119880)119881 (53)

we have

(nabla119883nabla119884119882lowast) (119885 119880)119881 = 119883119860 (119884) + 119860 (119883)119860 (119884)119882

lowast(119885 119880)119881

(54)

In view of (52) and (54) we have

(119877 (119883 119884) sdot 119882lowast) (119885 119880)119881 = [2119889119860 (119883 119884)]119882

lowast(119885 119880)119881

= 0

(55)

Thus by virtue of Theorem 3 the above equation shows that1198722119899+1 is an 120578-Einstein manifold This completes the proof

7 119873(120581)-Contact Metric Manifolds Satisfying119882lowastsdot 119878 = 0

Theorem8 If on an119873(120581)-contact metric manifold119882lowastsdot119878 = 0

then 119878(119876119883 119884) = 2120581[(119899 minus 1)119878(119883119884) + 2119899120581119892(119883 119884)]

Proof Let 119882lowast(120585 119883) sdot 119878 = 0 In this case we can write

119878 (119882lowast(120585 119883) 119884 119885) + 119878 (119884119882

lowast(120585 119883)119885) = 0 (56)

In view of (34) the above equation reduces to

minus 120581 [2119899120581 119892 (119883 119884) 120578 (119885) + 119892 (119883 119885) 120578 (119884)

minus 120578 (119884) 119878 (119883 119885) + 120578 (119885) 119878 (119883 119884)]

+1

2[2119899120581 119878 (119883 119884) 120578 (119885) + 119878 (119883 119885) 120578 (119884)

minus 120578 (119884) 119878 (119876119883119885) + 120578 (119885) 119878 (119876119883 119884)] = 0

(57)

Now putting 119885 = 120585 in above equation and using (3) (9) and(23) we get

119878 (119876119883 119884) = 2120581 [(119899 minus 1) 119878 (119883 119884) + 2119899120581119892 (119883 119884)] (58)

This completes the proof

8 119873(120581)-Contact Metric Manifolds Satisfying119882lowastsdot 119877 = 0

Theorem 9 On an119873(120581)-contact metric manifold if119882lowast sdot 119877 =

0 then 119878(119876119883 119884) = 2120581[(119899 minus 1)119878(119883 119884) + 2119899120581119892(119883 119884)]

Proof Suppose that 119882lowast(120585 119883) sdot 119877 = 0 then it can be writtenas

119882lowast(120585 119883) 119877 (119884 119885)119880 minus 119877 (119882

lowast(120585 119883) 119884 119885)119880

minus 119877 (119884119882lowast(120585 119883)119885)119880 minus 119877 (119884 119885)119882

lowast(120585 119883)119880 = 0

(59)

which on using (33) takes the form

minus120581

(119899 minus 1)[119892 (119883 119877 (119884 119885)119880) 120585 minus 120578 (119877 (119884 119885)119880)119883

minus 119892 (119883 119884) 119877 (120585 119885)119880 + 120578 (119884) 119877 (119883 119885)119880

minus 119892 (119883 119885) 119877 (119884 120585) 119880 + 120578 (119885) 119877 (119884119883)119880

minus119892 (119883119880) 119877 (119884 119885) 120585 + 120578 (119880) 119877 (119884 119885)119883]

minus1

2 (119899 minus 1)[1015840119877 (119884 119885119880 119876119883) 120585 minus 120578 (119877 (119884 119885)119880)119876119883

minus 119878 (119883 119884) 119877 (120585 119885)119880 + 120578 (119884) 119877 (119876119883119885)119880

minus 119878 (119883 119885) 119877 (119884 120585)119880 + 120578 (119885) 119877 (119884 119876119883)119880

minus119878 (119883119880) 119877 (119884 119885) 120585 + 120578 (119880) 119877 (119884 119885)119876119883]

= 0

(60)

Taking the inner product of above equation with 120585 we get

minus120581

(119899 minus 1)[1015840119877 (119884 119885 119880119883) minus 119892 (119883 119884) 120578 (119877 (120585 119885)119880)

+ 120578 (119884) 120578 (119877 (119883 119885)119880) minus 119892 (119883 119885) 120578 (119877 (119884 120585) 119880)

+ 120578 (119885) 120578 (119877 (119884119883)119880) minus 119892 (119883119880) 120578 (119877 (119884 119885) 120585)

+120578 (119880) 120578 (119877 (119884 119885)119883) ]

minus1

2 (119899 minus 1)[1015840119877 (119884 119885119880119876119883) minus 119878 (119883 119884) 120578 (119877 (120585 119885)119880)

+ 120578 (119884) 120578 (119877 (119876119883119885)119880) minus 119878 (119883 119885) 120578 (119877 (119884 120585)119880)

+ 120578 (119885) 120578 (119877 (119884 119876119883)119880) minus 119878 (119883119880) 120578 (119877 (119884 119885) 120585)

+120578 (119880) 120578 (119877 (119884 119885)119876119883) ] = 0

(61)

Now using (22) (28) and (29) in the above equation we get

minus120581

(119899 minus 1)[1015840119877 (119884 119885119880119883) + 120581 119892 (119883 119885) 119892 (119884 119880)

minus119892 (119883 119884) 119892 (119885119880) ]

6 ISRN Geometry

minus1

2 (119899 minus 1)[1015840119877 (119884 119885119880 119876119883) + 120581 119878 (119883 119885) 119892 (119884 119880)

minus119878 (119883 119884) 119892 (119885119880) ] = 0

(62)

Putting 119885 = 119880 = 119890119894 in the above equation and summing over119894 1 le 119894 le 2119899 + 1 we get

119878 (119876119883 119884) = 2120581 [(119899 minus 1) 119878 (119883 119884) + 2119899120581119892 (119883 119884)] (63)

This completes the proof

References

[1] S Tanno ldquoThe automorphism groups of almost contact Rie-mannian manifoldsrdquoThe Tohoku Mathematical Journal vol 21pp 21ndash38 1969

[2] K Kenmotsu ldquoA class of almost contact Riemannian mani-foldsrdquo The Tohoku Mathematical Journal vol 24 pp 93ndash1031972

[3] G P Pokhariyal and R S Mishra ldquoCurvature tensorsrsquo and theirrelativistics significancerdquo Yokohama Mathematical Journal vol18 pp 105ndash108 1970

[4] R H Ojha ldquoM-projectively flat Sasakian manifoldsrdquo IndianJournal of Pure and Applied Mathematics vol 17 no 4 pp 481ndash484 1986

[5] S K Chaubey and R H Ojha ldquoOn the m-projective curvaturetensor of a Kenmotsu manifoldrdquo Differential Geometry vol 12pp 52ndash60 2010

[6] R N Singh S K Pandey and G Pandey ldquoOn a type ofKenmotsu manifoldrdquo Bulletin of Mathematical Analysis andApplications vol 4 no 1 pp 117ndash132 2012

[7] J P Singh ldquoOn m-projective recurrent Riemannian manifoldrdquoInternational Journal ofMathematical Analysis vol 6 no 24 pp1173ndash1178 2012

[8] J-B Jun I B Kim and U K Kim ldquoOn 3-dimensional almostcontact metric manifoldsrdquo Kyungpook Mathematical Journalvol 34 no 2 pp 293ndash301 1994

[9] C Baikoussis D E Blair and T Koufogiorgos ldquoA decompo-sition of the curvature tensor of a contact manifold satisfying119877(119883 119884)120585 = 120581[120578(119884)119883minus120578(119883)119884]rdquo Mathematics Technical ReportUniversity of Ioanniana 1992

[10] B J Papantoniou ldquoContact Riemannian manifolds satisfying119877(120585119883)119877 = 0 and 120585 isin (120581 120583)-nullity distributionrdquo YokohamaMathematical Journal vol 40 no 2 pp 149ndash161 1993

[11] D E Blair T Koufogiorgos and B J Papantoniou ldquoContactmetric manifolds satisfying a nullity conditionrdquo Israel Journalof Mathematics vol 91 no 1ndash3 pp 189ndash214 1995

[12] E Boeckx ldquoA full classification of contact metric (120581 120583)-spacesrdquoIllinois Journal of Mathematics vol 44 no 1 pp 212ndash219 2000

[13] S Tanno ldquoRicci curvatures of contact Riemannian manifoldsrdquoThe Tohoku Mathematical Journal vol 40 no 3 pp 441ndash4481988

[14] D E Blair J-S Kim and M M Tripathi ldquoOn the concircularcurvature tensor of a contact metric manifoldrdquo Journal of theKorean Mathematical Society vol 42 no 5 pp 883ndash892 2005

[15] D E Blair T Koufogiorgos and R Sharma ldquoA classification of3-dimensional contact metric manifolds with119876120593 = 120593119876rdquo KodaiMathematical Journal vol 13 no 3 pp 391ndash401 1990

[16] D E Blair and H Chen ldquoA classification of 3-dimensionalcontact metric manifolds with 119876120593 = 120593119876 IIrdquo Bulletin of theInstitute of Mathematics vol 20 no 4 pp 379ndash383 1992

[17] U C De and A Sarkar ldquoOn a type of P-Sasakian manifoldsrdquoMathematical Reports vol 11(61) no 2 pp 139ndash144 2009

[18] G Zhen J L Cabrerizo L M Fernandez and M FernandezldquoOn 120585-conformally flat contact metric manifoldsrdquo Indian Jour-nal of Pure and AppliedMathematics vol 28 no 6 pp 725ndash7341997

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Discrete Dynamics in Nature and Society

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Decision SciencesAdvances in

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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

ISRN Geometry 3

where 119878 is the Ricci tensor of type (0 2) 119876 is the Riccioperator that is 119892(119876119883 119884) = 119878(119883 119884) and 119903 is the scalarcurvature of the manifold From (11) it follows that

(nabla119883120578) (119884) = 119892 (119883 + ℎ119883 120601119884) (17)

Also in a (120581 120583)-manifold

120578 (119877 (119883 119884)119885) = 120581 [119892 (119884 119885) 120578 (119883) minus 119892 (119883 119885) 120578 (119884)]

+ 120583 [119892 (ℎ119884 119885) 120578 (119883) minus 119892 (ℎ119883 119885) 120578 (119884)]

(18)

holdsThe 120581-nullity distribution119873(120581) of aRiemannianmanifold

1198722119899+1 [13] is defined by

119873(120581) 119901 997888rarr 119873119901 (120581) = 119885 isin 119879119901119872 119877 (119883 119884)119885

= 120581 (119892 (119884 119885)119883 minus 119892 (119883 119885) 119884)

(19)

for all119883119884 isin 119879119872 and 120581 being a constant If the characteristicvector field 120585 isin 119873(120581) then we call a contact metric manifoldan 119873(120581)-contact metric manifold [14] If 120581 = 1 then 119873(120581)-contact metric manifold is Sasakian and if 120581 = 0 then119873(120581)-contact metric manifold is locally isometric to the product119864119899+1

times 119878119899(4) for 119899 gt 1 and flat for 119899 = 1 If 120581 lt 1 the scalar

curvature is 119903 = 2119899(2119899 minus 2 + 120581) If 120583 = 0 then a (120581 120583)-contactmetric manifold reduces to a119873(120581)-contact metric manifoldsIn [9] 119873(120581)-contact metric manifold were studied in somedetail

In 119873(120581)-contact metric manifolds the following relationshold ([15 16])

ℎ2= (120581 minus 1) 120601

2 120581 le 1 (20)

(nabla119883120601) (119884) = 119892 (119883 + ℎ119883 119884) 120585 minus 120578 (119884) (119883 + ℎ119883) (21)

119877 (120585119883) 119884 = 120581 [119892 (119883 119884) 120585 minus 120578 (119884)119883] (22)

119878 (119883 120585) = 2119899120581120578 (119883) (23)

119878 (119883 119884) = 2 (119899 minus 1) [119892 (119883 119884) + 119892 (ℎ119883 119884)]

+ [2 (1 minus 119899) + 2119899120581] 120578 (119883) 120578 (119884) 119899 ge 1

(24)

119903 = 2119899 (2119899 minus 2 + 120581) (25)

119878 (120601119883 120601119884) = 119878 (119883 119884) minus 2119899120581120578 (119883) 120578 (119884)

minus 4 (119899 minus 1) 119892 (ℎ119883 119884)

(26)

(nabla119883120578) (119884) = 119892 (119883 + ℎ119883 120601119884) (27)

119877 (119883 119884) 120585 = 120581 [120578 (119884)119883 minus 120578 (119883)119884] (28)

120578 (119877 (119883 119884)119885) = 120581 [119892 (119884 119885) 120578 (119883) minus 119892 (119883 119885) 120578 (119884)] (29)

For a (2119899 + 1)-dimensional (119899 gt 1) almost contact metricmanifold m-projective curvature tensor 119882lowast is given by [3]

119882lowast(119883 119884)119885 = 119877 (119883 119884)119885 minus

1

2 (119899 minus 1)

times [119878 (119884 119885)119883 minus 119878 (119883 119885) 119884

+119892 (119884 119885)119876119883 minus 119892 (119883 119885)119876119884]

(30)

for arbitrary vector fields 119883 119884 and 119885 where 119878 is the Riccitensor of type (0 2) and 119876 is the Ricci operator that is119892(119876119883 119884) = 119878(119883 119884)

The m-projective curvature tensor 119882lowast for an 119873(120581)-

contact metric manifold is given by

119882lowast(119883 119884) 120585

= minus120581

(119899 minus 1)[120578 (119884)119883 minus 120578 (119883)119884]

minus1

2 (119899 minus 1)[120578 (119884)119876119883 minus 120578 (119883)119876119884]

(31)

120578 (119882lowast(119883 119884) 120585) = 0 (32)

119882lowast(120585 119884) 119885

= minus119882lowast(119884 120585) 119885

= minus120581

(119899 minus 1)[119892 (119884 119885) 120585 minus 120578 (119885) 119884]

minus1

2 (119899 minus 1)[119878 (119884 119885) 120585 minus 120578 (119885)119876119884]

(33)

120578 (119882lowast(120585 119884) 119885)

= minus120578 (119882lowast(119884 120585) 119885)

= minus120581

(119899 minus 1)[119892 (119884 119885) minus 120578 (119884) 120578 (119885)]

minus1

2 (119899 minus 1)[119878 (119884 119885) minus 2119899120581120578 (119884) 120578 (119885)]

(34)

120578 (119882lowast(119883 119884)119885)

= minus120581

(119899 minus 1)[119892 (119884 119885) 120578 (119883) minus 119892 (119883 119885) 120578 (119884)]

minus1

2 (119899 minus 1)[119878 (119884 119885) 120578 (119883) minus 119878 (119883 119885) 120578 (119884)]

(35)

3 M-Projectively Semisymmetric119873(120581)-Contact Metric Manifolds

Definition 1 A (2119899 + 1)-dimensional (119899 gt 1) 119873(120581)-contactmetric manifold is said to be m-projectively semisymmetric[17] if it satisfies 119877 sdot 119882

lowast= 0 where 119877 is the Riemannian

curvature tensor and119882lowast is them-projective curvature tensor

of the manifold

Theorem 2 An m-projectively semisymmetric 119873(120581)-contactmetric manifold is an Einstein manifold

4 ISRN Geometry

Proof Suppose that an 119873(120581)-contact metric manifold is m-projectively semisymmetric Then we have

(119877 (120585 119883) sdot 119882lowast) (119884 119885)119880 = 0 (36)

The above equation can be written as follows

119877 (120585119883)119882lowast(119884 119885)119880 minus 119882

lowast(119877 (120585 119883) 119884 119885)119880

minus 119882lowast(119884 119877 (120585 119883)119885)119880 minus 119882

lowast(119884 119885) 119877 (120585 119883)119880 = 0

(37)

In view of (22) the above equation reduces to

120581 [119892 (119883119882lowast(119884 119885)119880) 120585 minus 120578 (119882

lowast(119884 119885)119880)119883

minus 119892 (119883 119884)119882lowast(120585 119885)119880 + 120578 (119884)119882

lowast(119883 119885)119880

minus 119892 (119883 119885)119882lowast(119884 120585) 119880 + 120578 (119885)119882

lowast(119884119883)119880

minus119892 (119883119880)119882lowast(119884 119885) 120585 + 120578 (119880)119882

lowast(119884 119885)119883] = 0

(38)

Now taking the inner product of the above equation with 120585

and using (3) and (9) we get

120581 [1015840119882lowast(119884 119885 119880119883) minus 120578 (119882

lowast(119884 119885)119880) 120578 (119883)

minus 119892 (119883 119884) 120578 (119882lowast(120585 119885)119880) + 120578 (119884) 120578 (119882

lowast(119883 119885)119880)

minus 119892 (119883 119885) 120578 (119882lowast(119884 120585) 119880) + 120578 (119885) 120578 (119882

lowast(119884119883)119880)

minus119892 (119883119880) 120578 (119882lowast(119884 119885) 120585) + 120578 (119880) 120578 (119882

lowast(119884 119885)119883) ] = 0

(39)

which on using (30) (32) (34) and (35) gives

120581 [1015840119877 (119884 119885 119880119883) minus

1

2 (119899 minus 1)

times 119878 (119884119883) 119892 (119885119880) minus 119878 (119883 119885) 119892 (119884 119880)

+ 119878 (119883 119885) 120578 (119884) 120578 (119880) minus119878 (119883 119884) 120578 (119885) 120578 (119880)

+120581

(119899 minus 1)

times 119892 (119885119880) 119892 (119883 119884)

minus 119892 (119884119880) 119892 (119883 119885) + 119899119892 (119883 119885) 120578 (119884) 120578 (119880)

minus119899119892 (119883 119884) 120578 (119885) 120578 (119880) ] = 0

(40)

Putting 119885 = 119880 = 119890119894 in the above equation and takingsummation over 119894 1 le 119894 le 2119899 + 1 we get

119878 (119883 119884) = 2119899120581119892 (119883 119884) (41)

which shows that 1198722119899+1 is an Einstein manifold This com-pletes the proof

4 M-Projectively Flat 119873(120581)-ContactMetric Manifolds

Theorem 3 An m-projectively flat 119873(120581)-contact metric man-ifold 119872

2119899+1 is an Einstein manifold

Proof Let 119882lowast(119883 119884 119885 119880) = 0 Then from (30) we have1015840119877 (119883 119884 119885 119880)

=1

2 (119899 minus 1)[119878 (119884 119885) 119892 (119883119880)

minus 119878 (119883 119885) 119892 (119884 119880) + 119892 (119884 119885) 119878 (119883119880)

minus119892 (119883 119885) 119892 (119884 119880)]

(42)

Let 119890119894 be an orthonormal basis of the tangent space at anypoint Putting119884 = 119885 = 119890119894 in the above equation and summingover 119894 1 le 119894 le 2119899 + 1 we get

119878 (119883 119884) = minus119903119892 (119883 119884) (43)

which shows that 1198722119899+1 is an Einstein manifold This com-pletes the proof

5 120585-M-Projectively Flat 119873(120581)-ContactMetric Manifolds

Definition 4 A (2119899 + 1)-dimensional (119899 gt 1) 119873(120581)-contactmetric manifold is said to be 120585-m-projectively flat [18] if119882lowast(119883 119884)120585 = 0 for all 119883119884 isin 119879119872

Theorem 5 A (2119899 + 1)-dimensional (119899 gt 1) 119873(120581)-contactmetric manifold is 120585-m-projectively flat if and only if it is an120578-Einstein manifold

Proof Let 119882lowast(119883 119884)120585 = 0 Then in view if (30) we have

119877 (119883 119884) 120585 =1

2 (119899 minus 1)[119878 (119884 120585)119883 minus 119878 (119883 120585) 119884

+119892 (119884 120585) 119876119883 minus 119892 (119883 120585) 119876119884]

(44)

By virtue of (9) (23) and (28) the above equation reduces to1

2[120578 (119884)119876119883 minus 120578 (119883)119876119884] minus 120581 [120578 (119884)119883 minus 120578 (119883)119884] = 0 (45)

which by putting 119884 = 120585 gives119876119883 = 2120581 [minus119883 + (119899 + 1) 120578 (119883) 120585] (46)

Now taking the inner product of above equation with 119880 weget

119878 (119883119880) = 2120581 [minus119892 (119883119880) + (119899 + 1) 120578 (119883) 120578 (119880)] (47)which shows that 119873(120581)-contact metric manifold is an 120578-Einstein manifold Conversely suppose that (47) is satisfiedThen by virtue of (46) and (31) we have119882

lowast(119883 119884)120585 = 0This

completes the proof

6 M-Projectively Recurrent 119873(120581)-ContactMetric Manifolds

Definition 6 A nonflat Riemannian manifold 1198722119899+1 is said

to be m-projectively recurrent if its m-projective curvaturetensor 119882lowast satisfies the condition

nabla119882lowast= 119860 otimes 119882

lowast (48)

where 119860 is nonzero 1-form

ISRN Geometry 5

Theorem 7 If an 119873(120581)-contact metric manifold is m-projectively recurrent then it is an 120578-Einstein manifold

Proof We define a function 1198912

= 119892(119882lowast119882lowast) on 119872

2119899+1where the metric 119892 is extended to the inner product betweenthe tensor fields Then we have

119891 (119884119891) = 1198912119860 (119884) (49)

This can be written as

119884119891 = 119891 (119860 (119884)) (119891 = 0) (50)

From the above equation we have

119883(119884119891) minus 119884 (119883119891) = 119883119860 (119884) minus 119884119860 (119883) minus 119860 ([119883 119884]) 119891 (51)

Since the left-hand side of the above equation is identicallyzero and 119891 = 0 on 119872

2119899+1 then

119889119860 (119883 119884) = 0 (52)

that is 1-form 119860 is closedNow from

(nabla119884119882lowast) (119885 119880)119881 = 119860 (119884)119882

lowast(119885 119880)119881 (53)

we have

(nabla119883nabla119884119882lowast) (119885 119880)119881 = 119883119860 (119884) + 119860 (119883)119860 (119884)119882

lowast(119885 119880)119881

(54)

In view of (52) and (54) we have

(119877 (119883 119884) sdot 119882lowast) (119885 119880)119881 = [2119889119860 (119883 119884)]119882

lowast(119885 119880)119881

= 0

(55)

Thus by virtue of Theorem 3 the above equation shows that1198722119899+1 is an 120578-Einstein manifold This completes the proof

7 119873(120581)-Contact Metric Manifolds Satisfying119882lowastsdot 119878 = 0

Theorem8 If on an119873(120581)-contact metric manifold119882lowastsdot119878 = 0

then 119878(119876119883 119884) = 2120581[(119899 minus 1)119878(119883119884) + 2119899120581119892(119883 119884)]

Proof Let 119882lowast(120585 119883) sdot 119878 = 0 In this case we can write

119878 (119882lowast(120585 119883) 119884 119885) + 119878 (119884119882

lowast(120585 119883)119885) = 0 (56)

In view of (34) the above equation reduces to

minus 120581 [2119899120581 119892 (119883 119884) 120578 (119885) + 119892 (119883 119885) 120578 (119884)

minus 120578 (119884) 119878 (119883 119885) + 120578 (119885) 119878 (119883 119884)]

+1

2[2119899120581 119878 (119883 119884) 120578 (119885) + 119878 (119883 119885) 120578 (119884)

minus 120578 (119884) 119878 (119876119883119885) + 120578 (119885) 119878 (119876119883 119884)] = 0

(57)

Now putting 119885 = 120585 in above equation and using (3) (9) and(23) we get

119878 (119876119883 119884) = 2120581 [(119899 minus 1) 119878 (119883 119884) + 2119899120581119892 (119883 119884)] (58)

This completes the proof

8 119873(120581)-Contact Metric Manifolds Satisfying119882lowastsdot 119877 = 0

Theorem 9 On an119873(120581)-contact metric manifold if119882lowast sdot 119877 =

0 then 119878(119876119883 119884) = 2120581[(119899 minus 1)119878(119883 119884) + 2119899120581119892(119883 119884)]

Proof Suppose that 119882lowast(120585 119883) sdot 119877 = 0 then it can be writtenas

119882lowast(120585 119883) 119877 (119884 119885)119880 minus 119877 (119882

lowast(120585 119883) 119884 119885)119880

minus 119877 (119884119882lowast(120585 119883)119885)119880 minus 119877 (119884 119885)119882

lowast(120585 119883)119880 = 0

(59)

which on using (33) takes the form

minus120581

(119899 minus 1)[119892 (119883 119877 (119884 119885)119880) 120585 minus 120578 (119877 (119884 119885)119880)119883

minus 119892 (119883 119884) 119877 (120585 119885)119880 + 120578 (119884) 119877 (119883 119885)119880

minus 119892 (119883 119885) 119877 (119884 120585) 119880 + 120578 (119885) 119877 (119884119883)119880

minus119892 (119883119880) 119877 (119884 119885) 120585 + 120578 (119880) 119877 (119884 119885)119883]

minus1

2 (119899 minus 1)[1015840119877 (119884 119885119880 119876119883) 120585 minus 120578 (119877 (119884 119885)119880)119876119883

minus 119878 (119883 119884) 119877 (120585 119885)119880 + 120578 (119884) 119877 (119876119883119885)119880

minus 119878 (119883 119885) 119877 (119884 120585)119880 + 120578 (119885) 119877 (119884 119876119883)119880

minus119878 (119883119880) 119877 (119884 119885) 120585 + 120578 (119880) 119877 (119884 119885)119876119883]

= 0

(60)

Taking the inner product of above equation with 120585 we get

minus120581

(119899 minus 1)[1015840119877 (119884 119885 119880119883) minus 119892 (119883 119884) 120578 (119877 (120585 119885)119880)

+ 120578 (119884) 120578 (119877 (119883 119885)119880) minus 119892 (119883 119885) 120578 (119877 (119884 120585) 119880)

+ 120578 (119885) 120578 (119877 (119884119883)119880) minus 119892 (119883119880) 120578 (119877 (119884 119885) 120585)

+120578 (119880) 120578 (119877 (119884 119885)119883) ]

minus1

2 (119899 minus 1)[1015840119877 (119884 119885119880119876119883) minus 119878 (119883 119884) 120578 (119877 (120585 119885)119880)

+ 120578 (119884) 120578 (119877 (119876119883119885)119880) minus 119878 (119883 119885) 120578 (119877 (119884 120585)119880)

+ 120578 (119885) 120578 (119877 (119884 119876119883)119880) minus 119878 (119883119880) 120578 (119877 (119884 119885) 120585)

+120578 (119880) 120578 (119877 (119884 119885)119876119883) ] = 0

(61)

Now using (22) (28) and (29) in the above equation we get

minus120581

(119899 minus 1)[1015840119877 (119884 119885119880119883) + 120581 119892 (119883 119885) 119892 (119884 119880)

minus119892 (119883 119884) 119892 (119885119880) ]

6 ISRN Geometry

minus1

2 (119899 minus 1)[1015840119877 (119884 119885119880 119876119883) + 120581 119878 (119883 119885) 119892 (119884 119880)

minus119878 (119883 119884) 119892 (119885119880) ] = 0

(62)

Putting 119885 = 119880 = 119890119894 in the above equation and summing over119894 1 le 119894 le 2119899 + 1 we get

119878 (119876119883 119884) = 2120581 [(119899 minus 1) 119878 (119883 119884) + 2119899120581119892 (119883 119884)] (63)

This completes the proof

References

[1] S Tanno ldquoThe automorphism groups of almost contact Rie-mannian manifoldsrdquoThe Tohoku Mathematical Journal vol 21pp 21ndash38 1969

[2] K Kenmotsu ldquoA class of almost contact Riemannian mani-foldsrdquo The Tohoku Mathematical Journal vol 24 pp 93ndash1031972

[3] G P Pokhariyal and R S Mishra ldquoCurvature tensorsrsquo and theirrelativistics significancerdquo Yokohama Mathematical Journal vol18 pp 105ndash108 1970

[4] R H Ojha ldquoM-projectively flat Sasakian manifoldsrdquo IndianJournal of Pure and Applied Mathematics vol 17 no 4 pp 481ndash484 1986

[5] S K Chaubey and R H Ojha ldquoOn the m-projective curvaturetensor of a Kenmotsu manifoldrdquo Differential Geometry vol 12pp 52ndash60 2010

[6] R N Singh S K Pandey and G Pandey ldquoOn a type ofKenmotsu manifoldrdquo Bulletin of Mathematical Analysis andApplications vol 4 no 1 pp 117ndash132 2012

[7] J P Singh ldquoOn m-projective recurrent Riemannian manifoldrdquoInternational Journal ofMathematical Analysis vol 6 no 24 pp1173ndash1178 2012

[8] J-B Jun I B Kim and U K Kim ldquoOn 3-dimensional almostcontact metric manifoldsrdquo Kyungpook Mathematical Journalvol 34 no 2 pp 293ndash301 1994

[9] C Baikoussis D E Blair and T Koufogiorgos ldquoA decompo-sition of the curvature tensor of a contact manifold satisfying119877(119883 119884)120585 = 120581[120578(119884)119883minus120578(119883)119884]rdquo Mathematics Technical ReportUniversity of Ioanniana 1992

[10] B J Papantoniou ldquoContact Riemannian manifolds satisfying119877(120585119883)119877 = 0 and 120585 isin (120581 120583)-nullity distributionrdquo YokohamaMathematical Journal vol 40 no 2 pp 149ndash161 1993

[11] D E Blair T Koufogiorgos and B J Papantoniou ldquoContactmetric manifolds satisfying a nullity conditionrdquo Israel Journalof Mathematics vol 91 no 1ndash3 pp 189ndash214 1995

[12] E Boeckx ldquoA full classification of contact metric (120581 120583)-spacesrdquoIllinois Journal of Mathematics vol 44 no 1 pp 212ndash219 2000

[13] S Tanno ldquoRicci curvatures of contact Riemannian manifoldsrdquoThe Tohoku Mathematical Journal vol 40 no 3 pp 441ndash4481988

[14] D E Blair J-S Kim and M M Tripathi ldquoOn the concircularcurvature tensor of a contact metric manifoldrdquo Journal of theKorean Mathematical Society vol 42 no 5 pp 883ndash892 2005

[15] D E Blair T Koufogiorgos and R Sharma ldquoA classification of3-dimensional contact metric manifolds with119876120593 = 120593119876rdquo KodaiMathematical Journal vol 13 no 3 pp 391ndash401 1990

[16] D E Blair and H Chen ldquoA classification of 3-dimensionalcontact metric manifolds with 119876120593 = 120593119876 IIrdquo Bulletin of theInstitute of Mathematics vol 20 no 4 pp 379ndash383 1992

[17] U C De and A Sarkar ldquoOn a type of P-Sasakian manifoldsrdquoMathematical Reports vol 11(61) no 2 pp 139ndash144 2009

[18] G Zhen J L Cabrerizo L M Fernandez and M FernandezldquoOn 120585-conformally flat contact metric manifoldsrdquo Indian Jour-nal of Pure and AppliedMathematics vol 28 no 6 pp 725ndash7341997

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

4 ISRN Geometry

Proof Suppose that an 119873(120581)-contact metric manifold is m-projectively semisymmetric Then we have

(119877 (120585 119883) sdot 119882lowast) (119884 119885)119880 = 0 (36)

The above equation can be written as follows

119877 (120585119883)119882lowast(119884 119885)119880 minus 119882

lowast(119877 (120585 119883) 119884 119885)119880

minus 119882lowast(119884 119877 (120585 119883)119885)119880 minus 119882

lowast(119884 119885) 119877 (120585 119883)119880 = 0

(37)

In view of (22) the above equation reduces to

120581 [119892 (119883119882lowast(119884 119885)119880) 120585 minus 120578 (119882

lowast(119884 119885)119880)119883

minus 119892 (119883 119884)119882lowast(120585 119885)119880 + 120578 (119884)119882

lowast(119883 119885)119880

minus 119892 (119883 119885)119882lowast(119884 120585) 119880 + 120578 (119885)119882

lowast(119884119883)119880

minus119892 (119883119880)119882lowast(119884 119885) 120585 + 120578 (119880)119882

lowast(119884 119885)119883] = 0

(38)

Now taking the inner product of the above equation with 120585

and using (3) and (9) we get

120581 [1015840119882lowast(119884 119885 119880119883) minus 120578 (119882

lowast(119884 119885)119880) 120578 (119883)

minus 119892 (119883 119884) 120578 (119882lowast(120585 119885)119880) + 120578 (119884) 120578 (119882

lowast(119883 119885)119880)

minus 119892 (119883 119885) 120578 (119882lowast(119884 120585) 119880) + 120578 (119885) 120578 (119882

lowast(119884119883)119880)

minus119892 (119883119880) 120578 (119882lowast(119884 119885) 120585) + 120578 (119880) 120578 (119882

lowast(119884 119885)119883) ] = 0

(39)

which on using (30) (32) (34) and (35) gives

120581 [1015840119877 (119884 119885 119880119883) minus

1

2 (119899 minus 1)

times 119878 (119884119883) 119892 (119885119880) minus 119878 (119883 119885) 119892 (119884 119880)

+ 119878 (119883 119885) 120578 (119884) 120578 (119880) minus119878 (119883 119884) 120578 (119885) 120578 (119880)

+120581

(119899 minus 1)

times 119892 (119885119880) 119892 (119883 119884)

minus 119892 (119884119880) 119892 (119883 119885) + 119899119892 (119883 119885) 120578 (119884) 120578 (119880)

minus119899119892 (119883 119884) 120578 (119885) 120578 (119880) ] = 0

(40)

Putting 119885 = 119880 = 119890119894 in the above equation and takingsummation over 119894 1 le 119894 le 2119899 + 1 we get

119878 (119883 119884) = 2119899120581119892 (119883 119884) (41)

which shows that 1198722119899+1 is an Einstein manifold This com-pletes the proof

4 M-Projectively Flat 119873(120581)-ContactMetric Manifolds

Theorem 3 An m-projectively flat 119873(120581)-contact metric man-ifold 119872

2119899+1 is an Einstein manifold

Proof Let 119882lowast(119883 119884 119885 119880) = 0 Then from (30) we have1015840119877 (119883 119884 119885 119880)

=1

2 (119899 minus 1)[119878 (119884 119885) 119892 (119883119880)

minus 119878 (119883 119885) 119892 (119884 119880) + 119892 (119884 119885) 119878 (119883119880)

minus119892 (119883 119885) 119892 (119884 119880)]

(42)

Let 119890119894 be an orthonormal basis of the tangent space at anypoint Putting119884 = 119885 = 119890119894 in the above equation and summingover 119894 1 le 119894 le 2119899 + 1 we get

119878 (119883 119884) = minus119903119892 (119883 119884) (43)

which shows that 1198722119899+1 is an Einstein manifold This com-pletes the proof

5 120585-M-Projectively Flat 119873(120581)-ContactMetric Manifolds

Definition 4 A (2119899 + 1)-dimensional (119899 gt 1) 119873(120581)-contactmetric manifold is said to be 120585-m-projectively flat [18] if119882lowast(119883 119884)120585 = 0 for all 119883119884 isin 119879119872

Theorem 5 A (2119899 + 1)-dimensional (119899 gt 1) 119873(120581)-contactmetric manifold is 120585-m-projectively flat if and only if it is an120578-Einstein manifold

Proof Let 119882lowast(119883 119884)120585 = 0 Then in view if (30) we have

119877 (119883 119884) 120585 =1

2 (119899 minus 1)[119878 (119884 120585)119883 minus 119878 (119883 120585) 119884

+119892 (119884 120585) 119876119883 minus 119892 (119883 120585) 119876119884]

(44)

By virtue of (9) (23) and (28) the above equation reduces to1

2[120578 (119884)119876119883 minus 120578 (119883)119876119884] minus 120581 [120578 (119884)119883 minus 120578 (119883)119884] = 0 (45)

which by putting 119884 = 120585 gives119876119883 = 2120581 [minus119883 + (119899 + 1) 120578 (119883) 120585] (46)

Now taking the inner product of above equation with 119880 weget

119878 (119883119880) = 2120581 [minus119892 (119883119880) + (119899 + 1) 120578 (119883) 120578 (119880)] (47)which shows that 119873(120581)-contact metric manifold is an 120578-Einstein manifold Conversely suppose that (47) is satisfiedThen by virtue of (46) and (31) we have119882

lowast(119883 119884)120585 = 0This

completes the proof

6 M-Projectively Recurrent 119873(120581)-ContactMetric Manifolds

Definition 6 A nonflat Riemannian manifold 1198722119899+1 is said

to be m-projectively recurrent if its m-projective curvaturetensor 119882lowast satisfies the condition

nabla119882lowast= 119860 otimes 119882

lowast (48)

where 119860 is nonzero 1-form

ISRN Geometry 5

Theorem 7 If an 119873(120581)-contact metric manifold is m-projectively recurrent then it is an 120578-Einstein manifold

Proof We define a function 1198912

= 119892(119882lowast119882lowast) on 119872

2119899+1where the metric 119892 is extended to the inner product betweenthe tensor fields Then we have

119891 (119884119891) = 1198912119860 (119884) (49)

This can be written as

119884119891 = 119891 (119860 (119884)) (119891 = 0) (50)

From the above equation we have

119883(119884119891) minus 119884 (119883119891) = 119883119860 (119884) minus 119884119860 (119883) minus 119860 ([119883 119884]) 119891 (51)

Since the left-hand side of the above equation is identicallyzero and 119891 = 0 on 119872

2119899+1 then

119889119860 (119883 119884) = 0 (52)

that is 1-form 119860 is closedNow from

(nabla119884119882lowast) (119885 119880)119881 = 119860 (119884)119882

lowast(119885 119880)119881 (53)

we have

(nabla119883nabla119884119882lowast) (119885 119880)119881 = 119883119860 (119884) + 119860 (119883)119860 (119884)119882

lowast(119885 119880)119881

(54)

In view of (52) and (54) we have

(119877 (119883 119884) sdot 119882lowast) (119885 119880)119881 = [2119889119860 (119883 119884)]119882

lowast(119885 119880)119881

= 0

(55)

Thus by virtue of Theorem 3 the above equation shows that1198722119899+1 is an 120578-Einstein manifold This completes the proof

7 119873(120581)-Contact Metric Manifolds Satisfying119882lowastsdot 119878 = 0

Theorem8 If on an119873(120581)-contact metric manifold119882lowastsdot119878 = 0

then 119878(119876119883 119884) = 2120581[(119899 minus 1)119878(119883119884) + 2119899120581119892(119883 119884)]

Proof Let 119882lowast(120585 119883) sdot 119878 = 0 In this case we can write

119878 (119882lowast(120585 119883) 119884 119885) + 119878 (119884119882

lowast(120585 119883)119885) = 0 (56)

In view of (34) the above equation reduces to

minus 120581 [2119899120581 119892 (119883 119884) 120578 (119885) + 119892 (119883 119885) 120578 (119884)

minus 120578 (119884) 119878 (119883 119885) + 120578 (119885) 119878 (119883 119884)]

+1

2[2119899120581 119878 (119883 119884) 120578 (119885) + 119878 (119883 119885) 120578 (119884)

minus 120578 (119884) 119878 (119876119883119885) + 120578 (119885) 119878 (119876119883 119884)] = 0

(57)

Now putting 119885 = 120585 in above equation and using (3) (9) and(23) we get

119878 (119876119883 119884) = 2120581 [(119899 minus 1) 119878 (119883 119884) + 2119899120581119892 (119883 119884)] (58)

This completes the proof

8 119873(120581)-Contact Metric Manifolds Satisfying119882lowastsdot 119877 = 0

Theorem 9 On an119873(120581)-contact metric manifold if119882lowast sdot 119877 =

0 then 119878(119876119883 119884) = 2120581[(119899 minus 1)119878(119883 119884) + 2119899120581119892(119883 119884)]

Proof Suppose that 119882lowast(120585 119883) sdot 119877 = 0 then it can be writtenas

119882lowast(120585 119883) 119877 (119884 119885)119880 minus 119877 (119882

lowast(120585 119883) 119884 119885)119880

minus 119877 (119884119882lowast(120585 119883)119885)119880 minus 119877 (119884 119885)119882

lowast(120585 119883)119880 = 0

(59)

which on using (33) takes the form

minus120581

(119899 minus 1)[119892 (119883 119877 (119884 119885)119880) 120585 minus 120578 (119877 (119884 119885)119880)119883

minus 119892 (119883 119884) 119877 (120585 119885)119880 + 120578 (119884) 119877 (119883 119885)119880

minus 119892 (119883 119885) 119877 (119884 120585) 119880 + 120578 (119885) 119877 (119884119883)119880

minus119892 (119883119880) 119877 (119884 119885) 120585 + 120578 (119880) 119877 (119884 119885)119883]

minus1

2 (119899 minus 1)[1015840119877 (119884 119885119880 119876119883) 120585 minus 120578 (119877 (119884 119885)119880)119876119883

minus 119878 (119883 119884) 119877 (120585 119885)119880 + 120578 (119884) 119877 (119876119883119885)119880

minus 119878 (119883 119885) 119877 (119884 120585)119880 + 120578 (119885) 119877 (119884 119876119883)119880

minus119878 (119883119880) 119877 (119884 119885) 120585 + 120578 (119880) 119877 (119884 119885)119876119883]

= 0

(60)

Taking the inner product of above equation with 120585 we get

minus120581

(119899 minus 1)[1015840119877 (119884 119885 119880119883) minus 119892 (119883 119884) 120578 (119877 (120585 119885)119880)

+ 120578 (119884) 120578 (119877 (119883 119885)119880) minus 119892 (119883 119885) 120578 (119877 (119884 120585) 119880)

+ 120578 (119885) 120578 (119877 (119884119883)119880) minus 119892 (119883119880) 120578 (119877 (119884 119885) 120585)

+120578 (119880) 120578 (119877 (119884 119885)119883) ]

minus1

2 (119899 minus 1)[1015840119877 (119884 119885119880119876119883) minus 119878 (119883 119884) 120578 (119877 (120585 119885)119880)

+ 120578 (119884) 120578 (119877 (119876119883119885)119880) minus 119878 (119883 119885) 120578 (119877 (119884 120585)119880)

+ 120578 (119885) 120578 (119877 (119884 119876119883)119880) minus 119878 (119883119880) 120578 (119877 (119884 119885) 120585)

+120578 (119880) 120578 (119877 (119884 119885)119876119883) ] = 0

(61)

Now using (22) (28) and (29) in the above equation we get

minus120581

(119899 minus 1)[1015840119877 (119884 119885119880119883) + 120581 119892 (119883 119885) 119892 (119884 119880)

minus119892 (119883 119884) 119892 (119885119880) ]

6 ISRN Geometry

minus1

2 (119899 minus 1)[1015840119877 (119884 119885119880 119876119883) + 120581 119878 (119883 119885) 119892 (119884 119880)

minus119878 (119883 119884) 119892 (119885119880) ] = 0

(62)

Putting 119885 = 119880 = 119890119894 in the above equation and summing over119894 1 le 119894 le 2119899 + 1 we get

119878 (119876119883 119884) = 2120581 [(119899 minus 1) 119878 (119883 119884) + 2119899120581119892 (119883 119884)] (63)

This completes the proof

References

[1] S Tanno ldquoThe automorphism groups of almost contact Rie-mannian manifoldsrdquoThe Tohoku Mathematical Journal vol 21pp 21ndash38 1969

[2] K Kenmotsu ldquoA class of almost contact Riemannian mani-foldsrdquo The Tohoku Mathematical Journal vol 24 pp 93ndash1031972

[3] G P Pokhariyal and R S Mishra ldquoCurvature tensorsrsquo and theirrelativistics significancerdquo Yokohama Mathematical Journal vol18 pp 105ndash108 1970

[4] R H Ojha ldquoM-projectively flat Sasakian manifoldsrdquo IndianJournal of Pure and Applied Mathematics vol 17 no 4 pp 481ndash484 1986

[5] S K Chaubey and R H Ojha ldquoOn the m-projective curvaturetensor of a Kenmotsu manifoldrdquo Differential Geometry vol 12pp 52ndash60 2010

[6] R N Singh S K Pandey and G Pandey ldquoOn a type ofKenmotsu manifoldrdquo Bulletin of Mathematical Analysis andApplications vol 4 no 1 pp 117ndash132 2012

[7] J P Singh ldquoOn m-projective recurrent Riemannian manifoldrdquoInternational Journal ofMathematical Analysis vol 6 no 24 pp1173ndash1178 2012

[8] J-B Jun I B Kim and U K Kim ldquoOn 3-dimensional almostcontact metric manifoldsrdquo Kyungpook Mathematical Journalvol 34 no 2 pp 293ndash301 1994

[9] C Baikoussis D E Blair and T Koufogiorgos ldquoA decompo-sition of the curvature tensor of a contact manifold satisfying119877(119883 119884)120585 = 120581[120578(119884)119883minus120578(119883)119884]rdquo Mathematics Technical ReportUniversity of Ioanniana 1992

[10] B J Papantoniou ldquoContact Riemannian manifolds satisfying119877(120585119883)119877 = 0 and 120585 isin (120581 120583)-nullity distributionrdquo YokohamaMathematical Journal vol 40 no 2 pp 149ndash161 1993

[11] D E Blair T Koufogiorgos and B J Papantoniou ldquoContactmetric manifolds satisfying a nullity conditionrdquo Israel Journalof Mathematics vol 91 no 1ndash3 pp 189ndash214 1995

[12] E Boeckx ldquoA full classification of contact metric (120581 120583)-spacesrdquoIllinois Journal of Mathematics vol 44 no 1 pp 212ndash219 2000

[13] S Tanno ldquoRicci curvatures of contact Riemannian manifoldsrdquoThe Tohoku Mathematical Journal vol 40 no 3 pp 441ndash4481988

[14] D E Blair J-S Kim and M M Tripathi ldquoOn the concircularcurvature tensor of a contact metric manifoldrdquo Journal of theKorean Mathematical Society vol 42 no 5 pp 883ndash892 2005

[15] D E Blair T Koufogiorgos and R Sharma ldquoA classification of3-dimensional contact metric manifolds with119876120593 = 120593119876rdquo KodaiMathematical Journal vol 13 no 3 pp 391ndash401 1990

[16] D E Blair and H Chen ldquoA classification of 3-dimensionalcontact metric manifolds with 119876120593 = 120593119876 IIrdquo Bulletin of theInstitute of Mathematics vol 20 no 4 pp 379ndash383 1992

[17] U C De and A Sarkar ldquoOn a type of P-Sasakian manifoldsrdquoMathematical Reports vol 11(61) no 2 pp 139ndash144 2009

[18] G Zhen J L Cabrerizo L M Fernandez and M FernandezldquoOn 120585-conformally flat contact metric manifoldsrdquo Indian Jour-nal of Pure and AppliedMathematics vol 28 no 6 pp 725ndash7341997

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 5

Theorem 7 If an 119873(120581)-contact metric manifold is m-projectively recurrent then it is an 120578-Einstein manifold

Proof We define a function 1198912

= 119892(119882lowast119882lowast) on 119872

2119899+1where the metric 119892 is extended to the inner product betweenthe tensor fields Then we have

119891 (119884119891) = 1198912119860 (119884) (49)

This can be written as

119884119891 = 119891 (119860 (119884)) (119891 = 0) (50)

From the above equation we have

119883(119884119891) minus 119884 (119883119891) = 119883119860 (119884) minus 119884119860 (119883) minus 119860 ([119883 119884]) 119891 (51)

Since the left-hand side of the above equation is identicallyzero and 119891 = 0 on 119872

2119899+1 then

119889119860 (119883 119884) = 0 (52)

that is 1-form 119860 is closedNow from

(nabla119884119882lowast) (119885 119880)119881 = 119860 (119884)119882

lowast(119885 119880)119881 (53)

we have

(nabla119883nabla119884119882lowast) (119885 119880)119881 = 119883119860 (119884) + 119860 (119883)119860 (119884)119882

lowast(119885 119880)119881

(54)

In view of (52) and (54) we have

(119877 (119883 119884) sdot 119882lowast) (119885 119880)119881 = [2119889119860 (119883 119884)]119882

lowast(119885 119880)119881

= 0

(55)

Thus by virtue of Theorem 3 the above equation shows that1198722119899+1 is an 120578-Einstein manifold This completes the proof

7 119873(120581)-Contact Metric Manifolds Satisfying119882lowastsdot 119878 = 0

Theorem8 If on an119873(120581)-contact metric manifold119882lowastsdot119878 = 0

then 119878(119876119883 119884) = 2120581[(119899 minus 1)119878(119883119884) + 2119899120581119892(119883 119884)]

Proof Let 119882lowast(120585 119883) sdot 119878 = 0 In this case we can write

119878 (119882lowast(120585 119883) 119884 119885) + 119878 (119884119882

lowast(120585 119883)119885) = 0 (56)

In view of (34) the above equation reduces to

minus 120581 [2119899120581 119892 (119883 119884) 120578 (119885) + 119892 (119883 119885) 120578 (119884)

minus 120578 (119884) 119878 (119883 119885) + 120578 (119885) 119878 (119883 119884)]

+1

2[2119899120581 119878 (119883 119884) 120578 (119885) + 119878 (119883 119885) 120578 (119884)

minus 120578 (119884) 119878 (119876119883119885) + 120578 (119885) 119878 (119876119883 119884)] = 0

(57)

Now putting 119885 = 120585 in above equation and using (3) (9) and(23) we get

119878 (119876119883 119884) = 2120581 [(119899 minus 1) 119878 (119883 119884) + 2119899120581119892 (119883 119884)] (58)

This completes the proof

8 119873(120581)-Contact Metric Manifolds Satisfying119882lowastsdot 119877 = 0

Theorem 9 On an119873(120581)-contact metric manifold if119882lowast sdot 119877 =

0 then 119878(119876119883 119884) = 2120581[(119899 minus 1)119878(119883 119884) + 2119899120581119892(119883 119884)]

Proof Suppose that 119882lowast(120585 119883) sdot 119877 = 0 then it can be writtenas

119882lowast(120585 119883) 119877 (119884 119885)119880 minus 119877 (119882

lowast(120585 119883) 119884 119885)119880

minus 119877 (119884119882lowast(120585 119883)119885)119880 minus 119877 (119884 119885)119882

lowast(120585 119883)119880 = 0

(59)

which on using (33) takes the form

minus120581

(119899 minus 1)[119892 (119883 119877 (119884 119885)119880) 120585 minus 120578 (119877 (119884 119885)119880)119883

minus 119892 (119883 119884) 119877 (120585 119885)119880 + 120578 (119884) 119877 (119883 119885)119880

minus 119892 (119883 119885) 119877 (119884 120585) 119880 + 120578 (119885) 119877 (119884119883)119880

minus119892 (119883119880) 119877 (119884 119885) 120585 + 120578 (119880) 119877 (119884 119885)119883]

minus1

2 (119899 minus 1)[1015840119877 (119884 119885119880 119876119883) 120585 minus 120578 (119877 (119884 119885)119880)119876119883

minus 119878 (119883 119884) 119877 (120585 119885)119880 + 120578 (119884) 119877 (119876119883119885)119880

minus 119878 (119883 119885) 119877 (119884 120585)119880 + 120578 (119885) 119877 (119884 119876119883)119880

minus119878 (119883119880) 119877 (119884 119885) 120585 + 120578 (119880) 119877 (119884 119885)119876119883]

= 0

(60)

Taking the inner product of above equation with 120585 we get

minus120581

(119899 minus 1)[1015840119877 (119884 119885 119880119883) minus 119892 (119883 119884) 120578 (119877 (120585 119885)119880)

+ 120578 (119884) 120578 (119877 (119883 119885)119880) minus 119892 (119883 119885) 120578 (119877 (119884 120585) 119880)

+ 120578 (119885) 120578 (119877 (119884119883)119880) minus 119892 (119883119880) 120578 (119877 (119884 119885) 120585)

+120578 (119880) 120578 (119877 (119884 119885)119883) ]

minus1

2 (119899 minus 1)[1015840119877 (119884 119885119880119876119883) minus 119878 (119883 119884) 120578 (119877 (120585 119885)119880)

+ 120578 (119884) 120578 (119877 (119876119883119885)119880) minus 119878 (119883 119885) 120578 (119877 (119884 120585)119880)

+ 120578 (119885) 120578 (119877 (119884 119876119883)119880) minus 119878 (119883119880) 120578 (119877 (119884 119885) 120585)

+120578 (119880) 120578 (119877 (119884 119885)119876119883) ] = 0

(61)

Now using (22) (28) and (29) in the above equation we get

minus120581

(119899 minus 1)[1015840119877 (119884 119885119880119883) + 120581 119892 (119883 119885) 119892 (119884 119880)

minus119892 (119883 119884) 119892 (119885119880) ]

6 ISRN Geometry

minus1

2 (119899 minus 1)[1015840119877 (119884 119885119880 119876119883) + 120581 119878 (119883 119885) 119892 (119884 119880)

minus119878 (119883 119884) 119892 (119885119880) ] = 0

(62)

Putting 119885 = 119880 = 119890119894 in the above equation and summing over119894 1 le 119894 le 2119899 + 1 we get

119878 (119876119883 119884) = 2120581 [(119899 minus 1) 119878 (119883 119884) + 2119899120581119892 (119883 119884)] (63)

This completes the proof

References

[1] S Tanno ldquoThe automorphism groups of almost contact Rie-mannian manifoldsrdquoThe Tohoku Mathematical Journal vol 21pp 21ndash38 1969

[2] K Kenmotsu ldquoA class of almost contact Riemannian mani-foldsrdquo The Tohoku Mathematical Journal vol 24 pp 93ndash1031972

[3] G P Pokhariyal and R S Mishra ldquoCurvature tensorsrsquo and theirrelativistics significancerdquo Yokohama Mathematical Journal vol18 pp 105ndash108 1970

[4] R H Ojha ldquoM-projectively flat Sasakian manifoldsrdquo IndianJournal of Pure and Applied Mathematics vol 17 no 4 pp 481ndash484 1986

[5] S K Chaubey and R H Ojha ldquoOn the m-projective curvaturetensor of a Kenmotsu manifoldrdquo Differential Geometry vol 12pp 52ndash60 2010

[6] R N Singh S K Pandey and G Pandey ldquoOn a type ofKenmotsu manifoldrdquo Bulletin of Mathematical Analysis andApplications vol 4 no 1 pp 117ndash132 2012

[7] J P Singh ldquoOn m-projective recurrent Riemannian manifoldrdquoInternational Journal ofMathematical Analysis vol 6 no 24 pp1173ndash1178 2012

[8] J-B Jun I B Kim and U K Kim ldquoOn 3-dimensional almostcontact metric manifoldsrdquo Kyungpook Mathematical Journalvol 34 no 2 pp 293ndash301 1994

[9] C Baikoussis D E Blair and T Koufogiorgos ldquoA decompo-sition of the curvature tensor of a contact manifold satisfying119877(119883 119884)120585 = 120581[120578(119884)119883minus120578(119883)119884]rdquo Mathematics Technical ReportUniversity of Ioanniana 1992

[10] B J Papantoniou ldquoContact Riemannian manifolds satisfying119877(120585119883)119877 = 0 and 120585 isin (120581 120583)-nullity distributionrdquo YokohamaMathematical Journal vol 40 no 2 pp 149ndash161 1993

[11] D E Blair T Koufogiorgos and B J Papantoniou ldquoContactmetric manifolds satisfying a nullity conditionrdquo Israel Journalof Mathematics vol 91 no 1ndash3 pp 189ndash214 1995

[12] E Boeckx ldquoA full classification of contact metric (120581 120583)-spacesrdquoIllinois Journal of Mathematics vol 44 no 1 pp 212ndash219 2000

[13] S Tanno ldquoRicci curvatures of contact Riemannian manifoldsrdquoThe Tohoku Mathematical Journal vol 40 no 3 pp 441ndash4481988

[14] D E Blair J-S Kim and M M Tripathi ldquoOn the concircularcurvature tensor of a contact metric manifoldrdquo Journal of theKorean Mathematical Society vol 42 no 5 pp 883ndash892 2005

[15] D E Blair T Koufogiorgos and R Sharma ldquoA classification of3-dimensional contact metric manifolds with119876120593 = 120593119876rdquo KodaiMathematical Journal vol 13 no 3 pp 391ndash401 1990

[16] D E Blair and H Chen ldquoA classification of 3-dimensionalcontact metric manifolds with 119876120593 = 120593119876 IIrdquo Bulletin of theInstitute of Mathematics vol 20 no 4 pp 379ndash383 1992

[17] U C De and A Sarkar ldquoOn a type of P-Sasakian manifoldsrdquoMathematical Reports vol 11(61) no 2 pp 139ndash144 2009

[18] G Zhen J L Cabrerizo L M Fernandez and M FernandezldquoOn 120585-conformally flat contact metric manifoldsrdquo Indian Jour-nal of Pure and AppliedMathematics vol 28 no 6 pp 725ndash7341997

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

6 ISRN Geometry

minus1

2 (119899 minus 1)[1015840119877 (119884 119885119880 119876119883) + 120581 119878 (119883 119885) 119892 (119884 119880)

minus119878 (119883 119884) 119892 (119885119880) ] = 0

(62)

Putting 119885 = 119880 = 119890119894 in the above equation and summing over119894 1 le 119894 le 2119899 + 1 we get

119878 (119876119883 119884) = 2120581 [(119899 minus 1) 119878 (119883 119884) + 2119899120581119892 (119883 119884)] (63)

This completes the proof

References

[1] S Tanno ldquoThe automorphism groups of almost contact Rie-mannian manifoldsrdquoThe Tohoku Mathematical Journal vol 21pp 21ndash38 1969

[2] K Kenmotsu ldquoA class of almost contact Riemannian mani-foldsrdquo The Tohoku Mathematical Journal vol 24 pp 93ndash1031972

[3] G P Pokhariyal and R S Mishra ldquoCurvature tensorsrsquo and theirrelativistics significancerdquo Yokohama Mathematical Journal vol18 pp 105ndash108 1970

[4] R H Ojha ldquoM-projectively flat Sasakian manifoldsrdquo IndianJournal of Pure and Applied Mathematics vol 17 no 4 pp 481ndash484 1986

[5] S K Chaubey and R H Ojha ldquoOn the m-projective curvaturetensor of a Kenmotsu manifoldrdquo Differential Geometry vol 12pp 52ndash60 2010

[6] R N Singh S K Pandey and G Pandey ldquoOn a type ofKenmotsu manifoldrdquo Bulletin of Mathematical Analysis andApplications vol 4 no 1 pp 117ndash132 2012

[7] J P Singh ldquoOn m-projective recurrent Riemannian manifoldrdquoInternational Journal ofMathematical Analysis vol 6 no 24 pp1173ndash1178 2012

[8] J-B Jun I B Kim and U K Kim ldquoOn 3-dimensional almostcontact metric manifoldsrdquo Kyungpook Mathematical Journalvol 34 no 2 pp 293ndash301 1994

[9] C Baikoussis D E Blair and T Koufogiorgos ldquoA decompo-sition of the curvature tensor of a contact manifold satisfying119877(119883 119884)120585 = 120581[120578(119884)119883minus120578(119883)119884]rdquo Mathematics Technical ReportUniversity of Ioanniana 1992

[10] B J Papantoniou ldquoContact Riemannian manifolds satisfying119877(120585119883)119877 = 0 and 120585 isin (120581 120583)-nullity distributionrdquo YokohamaMathematical Journal vol 40 no 2 pp 149ndash161 1993

[11] D E Blair T Koufogiorgos and B J Papantoniou ldquoContactmetric manifolds satisfying a nullity conditionrdquo Israel Journalof Mathematics vol 91 no 1ndash3 pp 189ndash214 1995

[12] E Boeckx ldquoA full classification of contact metric (120581 120583)-spacesrdquoIllinois Journal of Mathematics vol 44 no 1 pp 212ndash219 2000

[13] S Tanno ldquoRicci curvatures of contact Riemannian manifoldsrdquoThe Tohoku Mathematical Journal vol 40 no 3 pp 441ndash4481988

[14] D E Blair J-S Kim and M M Tripathi ldquoOn the concircularcurvature tensor of a contact metric manifoldrdquo Journal of theKorean Mathematical Society vol 42 no 5 pp 883ndash892 2005

[15] D E Blair T Koufogiorgos and R Sharma ldquoA classification of3-dimensional contact metric manifolds with119876120593 = 120593119876rdquo KodaiMathematical Journal vol 13 no 3 pp 391ndash401 1990

[16] D E Blair and H Chen ldquoA classification of 3-dimensionalcontact metric manifolds with 119876120593 = 120593119876 IIrdquo Bulletin of theInstitute of Mathematics vol 20 no 4 pp 379ndash383 1992

[17] U C De and A Sarkar ldquoOn a type of P-Sasakian manifoldsrdquoMathematical Reports vol 11(61) no 2 pp 139ndash144 2009

[18] G Zhen J L Cabrerizo L M Fernandez and M FernandezldquoOn 120585-conformally flat contact metric manifoldsrdquo Indian Jour-nal of Pure and AppliedMathematics vol 28 no 6 pp 725ndash7341997

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