study on some transformations of riemann-finsler spaces manoj kumar
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STUDY ON SOMETRANSFORMATIONS OF
RIEMANN-FINSLER SPACES
THESIS SUBMITTED TO THEDST-CIMS
FACULTY OF SCIENCEIN PARTIAL FULFILMENT OF THE DEGREE OF
Doctor of Philosophyin
Mathematical Sciences
by
Manoj KumarEnrolment No. 341755
UNDER THE SUPERVISION OFDr. Bankteshwar Tiwari
Banaras Hindu UniversityOctober-2014
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Dedicated
To
My Beloved Parents
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COPYRIGHT c⃝, FACULTY OF SCIENCEBANARAS HINDU UNIVERSITY
VARANASI-221005, INDIA
2014.
ALL RIGHTS RESERVED.
.............................................................................
.............................................................................
DST-CENTRE FOR INTERDISCIPLINARY MATHEMATICAL SCIENCES
FACULTY OF SCIENCE
BANARAS HINDU UNIVERSITY
VARANASI-221005, INDIA.
UNDERTAKING FROM THE CANDIDATE
I hereby declare that I have completed the research work for the full time pe-
riod described under the clause VIII.(1) of the Ph.D. ordinance of the Banaras
Hindu University, Varanasi and that the research work embodied in this thesis
entitled “STUDY ON SOME TRANSFORMATIONS OF RIEMANN-
FINSLER SPACES” is my own work.
Date: Manoj Kumar
Place: Varanasi (Research Scholar)
CANDIDATE’S DECLARATION
I, Manoj Kumar certify that the work embodied in this Ph.D. thesis as “Study
on some transformations of Riemann-Finsler spaces ” is my own bonafide work
carried out by me under the supervision of Dr. Bankteshwar Tiwari for a
period from September 2011 to October 2014 at DST-CIMS, Banaras Hindu
University, Varanasi. The matter embodied in this Ph.D. thesis has not been
submitted for the award of any other degree/diploma.
I declare that I have faithfully acknowledged, given credit to and referred to
the research workers wherever their works have been cited in the text and the
body of the thesis. I further certify that I have not willfully lifted up some other’s
work, para, text, data, results, etc. reported in the journals, books, magazines,
reports, dissertations, thesis, etc., or available at web-sites and included them in
this Ph.D. thesis and cited as my own work.
Date: Manoj Kumar
Place: Varanasi (Research Scholar)
CERTIFICATE FROM THE SUPERVISOR
This is to certify that the above statement made by the candidate is correct to
the best of my knowledge.
Prof. Umesh Singh Dr. B. Tiwari
(Co-ordinator) (Supervisor)
DST-CENTRE FOR INTERDISCIPLINARY MATHEMATICAL SCIENCES
FACULTY OF SCIENCE
BANARAS HINDU UNIVERSITY
VARANASI- 221005.
COURSE WORK ANDPRE-SUBMISSION SEMINAR COMPLETION
CERTIFICATE
This is to certify that Mr. Manoj Kumar, a bonafide research scholar of
this centre, has successfully completed the Ph.D. course work and pre-submission
seminar requirement which is a part of his Ph.D. programme, on his thesis en-
titled, “STUDY ON SOME TRANSFORMATIONS OF RIEMANN-
FINSLER SPACES”.
Date: Prof. Umesh Singh
Place: Varanasi (Co-ordinator)
COPYRIGHT TRANSFER CERTIFICATE
Title of the Thesis: STUDY ON SOME TRANSFORMATIONS OF RIEMANN-
FINSLER SPACES.
Candidate’s Name: Manoj Kumar
Copyright Transfer
The undersigned hereby assigns to the Banaras Hindu University all rights under
copyright that may exist in and for the above thesis submitted for the award of
the Ph.D. degree.
Copyright c⃝, Faculty of Science
Banaras Hindu UniversityVaranasi-221005, INDIA
2014.
All rights reserved.
Manoj Kumar
Note: However, the author may reproduce or authorize others to reproduce ma-
terial extracted verbatim from the thesis or derivative of the thesis for author’s
personal use provided that the source and the University’s copyright notice are
indicated.
Acknowledgements
I am indebted to many people for their assistance and inspiration in this work.
First of all, I would like to express my sincere and deep gratitude to my supervisor;
Dr. Bankteshwar Tiwari, Associate Professor, DST-CIMS; for introducing this
interesting topic; his patience and guidance. He has always supported my Ph.D.
program with trust and enthusiasm, believing in my capabilities, doing all that
he could to improve my knowledge and sharing his passion for academic research
and teaching.
I am thankful to various Professors of B. H. U., in particular, Prof. Umesh
Singh (Co-ordinator, DST-CIMS), Prof. A. K. Shrivastava (Dean, Faculty of
Science and Additional Co-ordinator, DST-CIMS), Prof. R. S. Pathak (Ex-Head
and Professor, Deptt. of Mathematics, BHU), Prof. M. M. Tripathi (Deptt. of
Mathematics, BHU), Dr. M. K. Singh (DST-CIMS), Dr. R. Chaubey (DST-
CIMS) and Dr. S. K. Upadhayay (Deptt. of Mathematical Sciences, IIT, BHU)
for their valuable time and guidance during the Ph.D. program.
I would be failing my duty if I do not express my sincere thanks to Dr. Ab-
hishek Singh (Post doctoral fellow, DST-CIMS) for assisting with library searches
and other help.
All my work is a generous, wise and understanding critic. Our tireless and
efficient office assistant Manish Srivastava suffered through many drafts and was
able to guide me through the complexities of organizing the manuscript. I have
benefited from the helpful staff and the resources of the DST-CIMS, Faculty of
Science, B. H. U.
I am grateful to the DST-Centre for Interdisciplinary Mathematical Sciences
(CIMS) for awarding Junior Research Fellowship in march 2012 and Senior Re-
search Fellowship in march 2014. I relied heavily on the excellent supportive work
of my fellow Anjani Kumar Shukla, Dhram Raj Singh and Pranav Waila for ad-
vanced understanding of MS office, Latex, Matlab and Mathematica.
I am also conscious to suggestions made from the helpful elders researchers
and the resources of the DST-CIMS. It can not be expressed in words the helping
and cooperating nature of Dr.R. Chaubey (DST-CIMS) and Dr.Sapna Devi
(Deptt.of Mathematics, University of Allahabad, Allahabad) who motivated
me to do researches of interdisciplinary nature, specially related to other areas
Cosmology and Modeling, respectively.
A special thanks to my friends Dinesh Kumar, Ghanashyam Kr. Prajapati,
Ranadip Gangopadhyay, Ashutosh Singh of DST-CIMS and my Senior Vishal
Singh, Krishna Kumar Singh of Deptt. of Mathematics (BHU Varanasi), who
were kind enough to read a very rough first draft of the manuscript and offer
comments and advice.
Thanks to my old friends in Delhi and the new ones, who met along the way,
in Varanasi as well as all over India, during my Ph.D. period for their love and
support given me in hard times.
My thanks are also due to those who either directly or indirectly helped and
encouraged me for this work.
Last but not least, I owe a great debt of gratitude to my father Shri Harish
Chandra Verma and my mother Smt. Kanya Wati for their endless patience, love
and support throughout my schooling, for their blessing and encouraging me to
excel in every aspect of life, which always motivate me to choose a carrier in
research in Mathematical Sciences.
—Manoj Kumar
Preface
The object of this thesis is to study some transformations of Riemann-Finsler
spaces. For instance, Randers space may be treated as a little deformation of
a Riemannian space. More generally Finsler spaces with (α, β)- metrics may be
treated as a deformation of Riemannian space. To avoid referring to previous
knowledge of Riemann-Finsler Geometry, we include chapter 1, which contains
those concepts and results on Riemann-Finsler Geometry which are used in an
essential way in the rest of the book.
With this approach in mind, the thesis has been divided into Seven chapters.
Chapter 1 contains some definitions and literature survey relevant to the proposed
problems. A brief introduction of Riemann-Finsler geometry with some historical
development is given.
Thereafter in chapter 2, a special (α, β)- metric, which is considered as a
generalization of the Randers metric as well as of the Z. Shen’s square metric, has
been studied and the conditions for a Finsler space with this special metric to be
a Berwald space, a Douglas space and Weakly-Berwald space respectively, have
also been found.
In the chapter 3, we find a condition under which a Finsler space with
Randers change of m-th root metric is projectively related to a m-th root metric
and also we find a condition under which this Randers transformed m-th root
Finsler metric is locally dually flat. Moreover, if transformed Finsler metric is
conformal to the m-th root Finsler metric, then we prove that both of them reduce
to Riemannian metrics.
In chapter 4, we study the conformal transformation of m-th root Finsler
metric. The spray coefficients, Riemann curvatures and Ricci curvature of confor-
mally transformed m-th root metrics are shown to be certain rational functions of
direction. Further under certain conditions it is shown that a conformally trans-
formed m-th root metric is locally dually flat if and only if the transformation is
a homothety. Moreover the conditions for the transformed metrics to be Einstein
and Isotropic mean Berwald curvature are also found.
Chapter 5 is devoted to study the properties of a modified Finsler space
obtained by transformation of a Finsler space with the help of two normalised
semi-parallel vector fields.
The last two chapters are devoted to applications of Finsler spaces in Mathe-
matical Modelling and Mathematical Cosmology which shows the interdisciplinary
nature of our work.
In the second last chapter 6, the dynamics of a predator-prey model is pro-
posed and analyzed. Three types of refuges: those that protect a constant number
of prey population, a constant proportion of prey population and a function of
predator-prey encounters using refuges are considered. Linear stability analysis
based on Lyapunov theory and Jacobi stability analysis based on KCC theory are
carried out. Comparisons of results obtained in both cases shows that, Jacobi
stability analysis of these models reflects the better ecological interpretation.
In chapter 7, Finsler-Randers cosmological models in modified gravity the-
ories have been investigated. The de Sitter, power law and general exponential
solutions are assumed for the scale factor in the corresponding cosmological mod-
els. For each scenario, we have discussed all energy conditions in detail. We have
also investigated the behaviour of FR cosmological models in modified theories of
gravity like Einstein theory, Hoyle-Narlikar Creation field theory, Lyra geometry
and General class of scalar-tensor theories.
In the last a list of number of books and research papers on the subject is
given in References.
ii
Contents
Preface i
1 Introduction 1
1.1 Historical Development of Finsler Geometry . . . . . . . . . . . . . 1
1.2 The Geometry of Finsler Spaces . . . . . . . . . . . . . . . . . . . . 3
1.3 Review of Literature about Finsler Geometry . . . . . . . . . . . . 6
1.4 Connections and Covariant Differentiations . . . . . . . . . . . . . . 11
1.5 Flag curvature and S-curvature in Finsler geometry . . . . . . . . . 17
1.6 Some special Finsler spaces . . . . . . . . . . . . . . . . . . . . . . 19
1.7 KCC theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
1.8 Mathematical Cosmology in Finslerian space-time . . . . . . . . . . 22
2 On Finsler space with a special (α, β)-metric 24
2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
2.2 The condition to be a Berwald space . . . . . . . . . . . . . . . . . 27
2.3 The condition to be a Douglas space . . . . . . . . . . . . . . . . . 28
2.4 The condition to be a Weakly-Berwald space . . . . . . . . . . . . . 32
3 On Randers change of a Finsler space with m-th root metric 38
3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
3.2 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
3.3 Fundamental metric tensor of Randers transformed m-th root metric 41
3.4 Spray coefficients of Randers transformed m-th root metric . . . . 42
3.5 Conformally related Randers transformed m-th root metric . . . . 46
iii
3.6 Locally dually flatness of Randers transformed m-th root metric . . 49
4 On Conformal Transformation of m-th root Finsler metric 51
4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51
4.2 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52
4.3 Fundamental tensor and Spray coefficients of conformally trans-
formed m-th root metric . . . . . . . . . . . . . . . . . . . . . . . . 53
4.4 Locally dually flat conformally transformed m-th root metric . . . . 55
4.5 Conformally transformed Einstein m-th root metric . . . . . . . . . 57
4.6 Conformally transformed m-th root metric with Isotropic E-curvature 58
5 Transformation of a Finsler Space by Normalised Semi-Parallel
Vector Fields 60
5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60
5.2 Transformed Finsler space obtained by Normalised semi-parallel
vector fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62
5.3 Special Finsler spaces with semi-parallel vector fields . . . . . . . . 65
6 Predator-prey model with prey refuges: Jacobi stability vs Linear
stability 68
6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68
6.2 Preliminary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70
6.3 Applications of geometric theory to second order system . . . . . . 72
6.4 Mathematical models . . . . . . . . . . . . . . . . . . . . . . . . . . 74
6.5 Numerical simulations and discussion . . . . . . . . . . . . . . . . . 78
6.6 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86
7 Finsler-Randers Cosmological models in Modified Gravity Theo-
ries 87
7.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87
7.2 Finsler-Randers Cosmological Model in Einstein Theory . . . . . . . 89
7.3 Finsler-Randers cosmological model in Lyra geometry . . . . . . . . 97
iv
7.4 Finsler-Randers Cosmological Model in General Class of Scalar-
tensor theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103
7.5 Finsler-Randers Cosmological Model in C-field theory . . . . . . . . 109
7.6 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114
Bibliography 116
v
Chapter 1
Introduction
1.1 Historical Development of Finsler Geometry
The differential geometry of Non-Euclidean spaces was studied systematically by
German mathematician B. Riemann (1854). He proposed metrics which are more
general than Euclidean. One metric, which is a quadratic differential one form,
is called Riemannian metric. Einstein’s general theory of relativity (1915) was its
first important and fully successful application. In the renowned lecture of 1854 ‘
Uber die Hypothesen, welcheder Geometrie zu Grunde liegen’ (On the Hypotheses,
which lie at the Foundations of Geometry), Riemann had himself conjectured the
existence of a geometry which is more general than the one proposed by him.
More than sixty years after his lecture a systematic study of generalized geometry,
began after the publication of a thesis by P. Finsler (1894-1970) in 1918, called
Finsler geometry. Finsler geometry has started with Finsler’s famous dissertation
under the supervision of C. Caratheodory who intended to geometrize the calculus
of variations.
The calculus of variations with specific contact to other geometrical back-
ground, are closely related by Finsler geometry and which was introduced by
Caratheodory in connection with problems in parametric form. In fact, the noti-
fiable relationship between some contact of differential geometry and the calculus
of variations have been seen some years ago to the publication of Finsler’s thesis,
in particular by Bliss (1906), Landsberg (1908) and Blaschke (1923). Definitions
of angle in terms of invariants of a parametric problem in the calculus of varia-
tion was introduced by Bliss and Landsberg. Although the geometrical theories
1
given by Bliss and Landsberg were against an Euclidean background, which were
not supporting the true objectives of the generalization of Riemann’s proposal.
Finsler’s thesis must be regarded as the first step in this direction.
In the year 1900 in Paris, the problem of geometry of the calculus of variations
was the one of the problems formulated by Hilbert. This problem was posed by
Hilbert without any formulation of specific question or conjecture. By providing
an account to the invariant integral, as well as emphasizing on the importance of
the problem of multiple integrals, Hilbert praised Kneser’s book also.
In all modern treatments of the subject, the role of Hilbert invariant integral
is crucial one. The geometrical idea in Finsler geometry is of a smoothly varying
family of Minkowski norms, rather than a family of inner products and this family
of Minkowski norms is known as a Finsler structure. In contrast to Riemannian ge-
ometry, there is also an equivalence problem that states that ‘how transformation
induced from a coordinate change can affect two given Finsler structures’? It is
quite reasonable to expect that a connection and its curvature, will be involved in
the solution of the equivalence problem in permissible set up of Finsler geometry.
A Finsler manifold is a manifold where each tangent space is equipped with
a Minkowski norm, that is, a norm that is not necessarily induced by an inner
product. This norm also induces a canonical inner product. However, in sharp
contrast to the Riemannian case, these Finsler-inner product does not only depend
on where you are, but also in which direction you are looking. Despite this quite
large step away from Riemannian geometry, Finsler geometry contains analogues
for many of the natural objects in Riemannian geometry. For example, length,
geodesics, curvature, connections, covariant derivative, and structure equations.
The history of development of Finsler geometry can be divided into four periods.
The first period of the history of development of Finsler geometry began in
1924, when three geometers J. H. Taylor, J. L. Synge and L. Berwald simulta-
neously started work in this field. However according to Matsumoto the creator
of this geometry is L. Berwald in 1925. The name “Finsler geometry” was first
given by J. Taylor in 1927.
The second period began in 1934, when E. Cartan published his thesis on
2
Finsler geometry. He showed that is was indeed possible to define connection
coefficients and hence covariant derivatives such that the Ricci lemma is satisfied.
On this basis Cartan developed the theory of curvature and torsion. All subsequent
investigations considering the geometry of Finsler spaces were dominated by this
approach. Several Mathematicians such as E. T. Davies, S. Golab, O. Varga, V.
V. Wagner have studied Finsler geometry along Cartan’s approach.
The third period of the history of Finsler geometry began in 1951, when H.
Rund introduced a new process of parallelism from the stand-point of Minkowskian
geometry. Cartan had introduced parallelism from the stand-point of locally Eu-
clidean geometry. Later on E. T. Davies and A. Diecke showed that the two con-
cepts of parallelism were the same. Several Mathematicians such as W. Barthel,
A. Diecke, D. Laugwitz have studied Finsler spaces on Rund’s approach.
The fourth period of the history of developmental of Finsler geometry began
in 1963, where H. Akabar developed the modern theory of Finsler spaces based
on the geometry of connections of fibres bundles. The reason of modernization is
to establish a global definition of connections in Finsler spaces and to re-examine
Cartan’s system of axioms. Mathematicians and Physicists began to study special
Finsler spaces since Matsumoto organised Symposium on the models of Finsler
spaces in 1970.
1.2 The Geometry of Finsler Spaces
The goal which Riemann set for himself was the definition and discussion of the
most general finite-dimensional space in which every curve has a length derived
from an infinitesimal length or line element. In modern terminology Riemann’s
approach is this: Let a differentiable manifold M of a certain class be given. In
any local coordinate system (x1, ..., xn) = (x) a length F (x, dx) must be assigned
to a given line element (x, dx) = (x1, ..., xn; dx1, ..., dxn) with origin x. If x(t) is a
(smooth) curve in M then∫F (x, x)dt is its length.
In order to insure that the length of a curve is positive and independent of the
3
sense in which the curve is traversed, Riemann requires F (x, dx) > 0 for dx = 0
and F (x, dx) = F (x,−dx).
Next Riemann assumes that the length of the line element remains unchanged
except for terms of second order, if all points undergo the same infinitesimal
change. This amounts to the condition F (x, kdx) = kF (x, dx) for k > 0. Now a
days we rather justify this condition by requiring that a change of the parametriza-
tion of the curve does not change its length. Riemann then turns immediately to
the special case where F (x, dx) =√∑
gij(x)dxidxj, that is, to those spaces which
are now called Riemann spaces. The general case is passed over with the following
remarks: the next simplest case would comprise the manifolds, in which the line
element can be expressed as the fourth root of a bi quadratic differential form.
The investigation of these more general types would not require any essentially
different principles, but it would be time consuming and contribute comparatively
little new to the theory of space (verhaltnismassig auf die Lehre vom Raume wenig
neues Licht werfen), because the results cannot be interpreted geometrically.
Here is one of the few instances where Riemann’s feeling was wrong. Nev-
ertheless the passage had a great influence: the general case was for along time
entirely neglected, and when it was taken up the principles of Riemannian geom-
etry were applied. The results thus obtained are not different enough to enrich
geometry materially, moreover they frequently do not lend themselves to a naive
geometric interpretation.
Finsler was the first who studied the general spaces systematically in his thesis
written under Caratheodory’s guidance. His main idea is this: if a curve is given,
then a field which contains the curve defines a Riemann metric as above. Those
results which are independent of the choice of the field are the real geometric
properties of the curve. In this way Finsler succeeded in developing a theory of
curves, and also the foundations for a theory of surfaces. This is the reason why
the name Finsler space was generally accepted for the general class of spaces first
defined by Riemann.
Then a new line of thought developed in the geometric school at Prague with
Berwald, Funk, and Winternitz as principle representatives. A Finsler space is
4
not considered as a point space but primarily as a set of line elements in which a
Riemannian metric is associated with each line element. The main emphasis in this
theory lies on the definitions and properties of parallelism and similar questions.
The whole development culminated in Cartan’s monograph, which is considered
to have given the theory of Finsler spaces its final form.
Under the restriction to use nothing but Riemannian methods it may be true
that the theory has reached its limits. However, as soon as this restriction is
dropped it appears that the surface has hardly been scratched.
A first, but not the most interesting or decisive, step beyond the mentioned
results consists in extending the results of Riemannian geometry rather than its
methods.
Finsler geometry is an alternative approach to geometrization of fields, and
its fundamental idea can be traced back to a lecture of Riemann (1854). In this
lecture Riemann suggested that the positive fourth root of a fourth order differen-
tial form might serve as a metric function. This function has three properties that
it is convex and common with the Riemannian quadratic form it is positive and
homogeneous of degree one in the differentials. Therefore, it is a natural general-
ization of the notion of distance between two neighbouring points xi; xi + dxi to
consider as given by some function F (xi; dxi), where i = 0, ..., n, satisfying these
three properties.
A systematic study of these kind of manifolds was first considered by Funk
(1929), and in 1925, the method of tensor calculus were applied to the theory of
Berwald (1947) independently but almost simultaneously by Synge, Taylor and
Berwald. It was found that the second derivatives of 12F 2(xi, dxi) with respect
to differentials serves as components of a metric tensor in analogy with Riemann
geometry. By this process, parallel displacements and connection coefficients in
Finsler spaces are defined, but with these connections Ricci lemma was no longer
valid. Cartan (1934) showed that it was indeed possible to define connections
and a covariant derivative so that Ricci lemma is preserved. This development is
closely related to the present application of Finsler geometry in physics, namely,
to geometrize both Cosmology and gravity simultaneously.
5
Finsler geometry was first applied in gravitational theory, and this application
lead to corrections to observational results predicted by general relativity.
As mentioned before, the main application of Finsler geometry is the ge-
ometrization of electromagnetism and gravitation. A Finslerian approach to this
geometrization was first introduced by Randers (1941), but in his work Finsler
geometry was not mentioned, although it was used. Randers metric produces a
geodesic equation identical with Lorentz equation for a charged particle. But the
metric depends on qm
and defines a different space for each type of particle by
Ingarden (1976).
Some (α, β)- metrics are important for Cosmology, in application point of
view. Presently Finsler geometry has many developments which focus to the
researchers. Now days Finsler geometry has found an abundance of applications
in both physics and practical applications.
1.3 Review of Literature about Finsler Geometry
Finsler geometry has been studied and developed by geometers like J. H. Taylor,
L. Berwald, E. Cartan, E. T. Davis, O. Varga, A. Deicke, M. Matsumoto, H.
Shimada, S. S. Chern, D. Bao, Z. Shen, L. Tamassy and R. Miron etc. This
geometry has the crucial applications due to G. S. Asanov, P. L. Antonelli, R. G.
Beil, R. M. Santilli, and R. S. Ingarden, etc.
1.3.1 Differentiable Manifold
An n-dimensitional differentiable manifold is a set M together with a family of
injective maps fi : Ui ⊂ Rn → fi(Ui) ⊆M of open sets Ui in Rn into M such that
(i) ∪ifi(Ui) =M ,
(ii) For each pair i, j with fi(Ui) ∩ fj(Uj) = W = ϕ,
the sets f−1i (W ) and f−1
j (W ) are open sets in Rn and f−1i ofj, f−1
j ofi are differen-
tiable.
(iii) The family (Ui, fi) is maximal relative to (i) and (ii).
Examples.
6
1. Rn is an n-dimensitional differentiable manifold.
2. Let Sn be the standard unit sphere in Rn+1 defines as
Sn :=
ξ = (ξi) ∈ Rn+1 : |ξ| =
√n+1∑i=1
(ξi)2 = 1
is n-dimensitional differentiable
manifold.
1.3.2 Minkowski Norm
Definition 1.1 A Minkowski norm on a finite dimensional (n−dim) vector
space V is a function F : V → [0,∞) such that
(i) F (y) ≥ 0 for any y ∈ V and F (y) = 0 if and only if y = 0,
(ii) F is positively 1-homogeneous, i.e. F (λy) = λF (y), for every y ∈ V and
λ > 0,
(iii) F is C∞ on V \0 such that for any y ∈ V , the bilinear symmetric functional
gy on V , defined as gy(u, v) := 12
∂2
∂s∂t[F 2(y + su + tv)]s=t=0, is an inner product,
i.e. for a fixed basis bi of V and y = yibi = 0 the matrix (gij(y)) defined by
gij(y) := gy(bi, bj) =12
∂2
∂yi∂yj[F 2] (y) should be positive definite.
The inner product gy is called fundamental form in the y direction. The pair
(V, F ) is called a Minkowski space. A Minkowski norm is said to be reversible if
F (−y) = F (y) for all y ∈ V .
Examples
1. Let V = Rn and y = (yi) ∈ Rn, |y| :=√
n∑i=1
(yi)2 be standard Euclidean norm,
then F (y) = |y| is a Minkowski norm and (Rn, F ) is a Minkowski space.
2. Let V = R2 and for y = (y1, y2) ∈ R2, let F (y) = ((y1)4 + (y2)4)14 , called quar-
tic norm. This F is NOT a Minkowski norm on R2, because det(gij) = 3(y1)2(y2)2
(y1)4+(y2)4
vanishes on co-ordinate axes and hence positive definiteness of matrix (gij) is vi-
olated at some nonzero y.
3. Let V = R2 and for y = (y1, y2) ∈ R2, let F (y) =√√
(y1)4 + (y2)4 + λ [(y1)2 + (y2)2];
this may be viewed as a perturbation of the quartic norm. This F is a Minkowski
norm on R2 for positive λ.
7
1.3.3 Finsler space
Let M be an n-dimensional C∞-manifold. TxM denotes the tangent space of M
at x. The tangent bundle of M is the union of tangent spaces TM :=∪
x∈M TxM .
We denote the elements of TM by (x, y) where y ∈ TxM . Let TM0 = TM \ 0
is called slit tangent bundle.
Definition 1.2 A Finsler metric on M is a function F : TM → [0,∞) with
the following properties:
(i) F is C∞ on slit tangent bundle TM0,
(ii) F is positively 1-homogeneous on the fibers of tangent bundle TM ,
and (iii) the Hessian of F 2
2with element gij(x, y) = 1
2∂2F 2
∂yi∂yjis positive definite on
TM0.
The pair (M,F ) is then called a Finsler space. A Finsler metric F = F (x, y) on a
manifold is said to be reversible if F (x,−y) = F (x, y) for all y ∈ TxM . Normally,
one does not impose this reversibility condition on a Finsler metric, because it
excludes some interesting examples such as a Randers metric.
A Finsler metric F on M is said to be Riemannian, if the restriction of F , Fx(y) :=
F (x, y) on TxM is a Euclidean norm for any x ∈ M ; that is, Fx(y) =√⟨y, y⟩
x,
for any y ∈ TxM , where ⟨, ⟩x is an inner product on TxM . One usually denotes a
Riemannian metric by a family of inner products gx = ⟨y, y⟩x, on tangent spaces
TxM .
The Riemannian metrics, which are reversible Finsler metrics, are important ex-
amples of Finsler metrics.
Examples [Tiwari (2012a)]
1. Euclidean metric
Let |y| :=
√n∑
i=1
(yi)2, y = (yi) ∈ Rn be the standard Euclidean norm on Rn.
Considering the identification TxRn ∼= Rn, define α0 = α0(x, y) by α0 := |y|, for
y ∈ Rn, then α0 is a Finsler metric called the standard Euclidean metric.
2. Spherical metric
Let Sn be the standard unit sphere in Rn+1 defines as
8
Sn :=
ξ = (ξi) ∈ Rn+1 : |ξ| =
√n+1∑i=1
(ξi)2 = 1
.
Then spherical metric is given by
α+1 := α+1(y) =
√|y|2 + (|x|2|y|2 − ⟨x, y⟩2)
1 + |x|2,
where notation ⟨, ⟩ denotes usual Euclidean inner product on Rn.
3. Klein Hyperbolic metric
Let Hn+ be upper portion of hyperboloid of two sheets given by
Hn := ξ = (ξi) ∈ Rn+1 : −(ξ1)2 − (ξ2)2 − ...− (ξn)2 + (ξn+1)2 = 1.
Then Klein Hyperbolic metric is given by
α−1 := α−1(y) =
√|y|2 − (|x|2|y|2 − ⟨x, y⟩2)
1− |x|2.
The three metrics in Examples (1), (2) and (3) can be incorporated in a single
formula:
αµ =
√|y|2 + µ(|x|2|y|2 − ⟨x, y⟩2)
1 + µ|x|2, y ∈ TxDn(rµ) ∼= Rn,
where rµ := 1√−µ
if µ < 0 and rµ := +∞ if µ ≥ 0.
4. Poincare Hyperbolic metric
We can construct Poincare Hyperbolic metric through stereographic projection of
the upper portion of the hyperboloid
HnR := ξ = (ξi) ∈ Rn+1 : −(ξ1)2 − (ξ2)2 − ...− (ξn)2 + (ξn+1)2 = R2
onto the ball Dn2R := x ∈ Rn : |x| < 2R, as follows
F = 4R2 |y|4R2 − |x|2
.
5. Funk metric
Let
Θ :=
√|y|2−(|x|2|y|2−⟨x,y⟩2)+⟨x,y⟩
1−|x|2 , y ∈ TxDn ∼= Rn.
Then Θ = Θ(x, y) is a Finsler metric on Dn, called the funk metric on Dn.
6. Berwald metric
Let
B :=
(√|y|2−(|x|2|y|2−⟨x,y⟩2)+⟨x,y⟩
)2
(1−|x|2)2√
|y|2−(|x|2|y|2−⟨x,y⟩2),
where y ∈ TxDn ∼= Rn. Then B = B(x, y) is a Finsler metric on Dn, first con-
structed by L. Berwald.
9
7. Randers metric
This metric was first introduced by a physicist named Randers (1941), who was
concerned with the unified field theory of gravitation and electromagnetism. His
justification was: ‘Perhaps the most characteristic property of the physical world is
the uni direction of time-like intervals. Since there is no reason why this asymme-
try should disappear in the mathematical description, it is of interest to consider
the possibility of a metric with asymmetrical property’. Later on, Ingarden (1957)
also used this metric in the context of electron microscopes and called it the Ran-
ders metric, for the first time, taking the name of G. Randers. Let α =√aij(x)yiyj
is a Riemannian metric and β = bi(x)yi is a one-form. Then the metric F = α+β
is a Finsler metric, called Randers metric.
8. (α, β)-metrics
M. Matsumoto has introduced the concept of (α, β)- metric in 1972 as a general-
ization of Randers metric.
A Finsler metric F on a differentiable manifold M is called an (α, β)-metric, where
α is a Riemannian metric α =√aij(x)yiyj and β is a one-form β = bi(x)y
i, if
F is a positively homogeneous function of degree one in α and β. The study of
some important (α, β) metrics such as Randers metric F = α+β, Kropina metric
F = α2
β, Matsumoto metric F = α2
α−β, generalized Kropina metric F = αn+1
βn and Z.
Shen’s square metric F = (α+β)2
αhave greatly contributed to the growth of Finsler
geometry and its applications to the theory of relativity and Cosmology and other
areas.
An (α, β)-metric can be expressed in the form F = αϕ(s) where ϕ is C∞ posi-
tive function and s = βα. F = αϕ(s) is a Finsler metric for any α and β with
b = ||β||α < b0 if and only if
ϕ(s) > 0, ϕ(s)− sϕ′(s) + (b2 − s2)ϕ′′ (s) > 0 , (|s| ≤ b < b0).
9. m-th root metrics
The concept of m-th root metric was introduced by Shimada (1979) and is given
by F := ai1i2...im(x)yi1yi2 ...yim1m , where ai1...im is a symmetric covariant tensor
of order m. The m-th root metric has been successfully applied to ecology by An-
tonelli et al. (1993) and studied by several authors [Srivastava and Arora (2012),
10
Tayebi et al. (2012), Tayebi and Najafi (2011), Yu and You (2010) and Prasad and
Dwivedi (2002)]. It is regarded as a generalization of Riemannian metric in the
sense that the second root metric is a Riemannian metric. For m = 3, it is called
a cubic Finsler metric and for m = 4, it is quartic metric. In four-dimension, the
special fourth root metric in the form F = 4√y1y2y3y4 is called the Berwald-Moor
metric.
1.3.4 Geodesics and Spray coefficients
A smooth curve (C∞) in a Finsler space is a geodesic if it has constant speed and
is locally minimizing. Thus a geodesic in (M,F ) is a curve c : I = [a, b] → M
with F (c(t), c(t)) = constant and for any t0 ∈ I, there is a small number ϵ > 0
such that c is minimizing on [t0 − ϵ, t0 + ϵ] ∩ I.
Using the calculus of variations, one can show that geodesics in a Finsler space F n
are given by a system of second order ordinary differential equations. If a geodesic
is represented locally by the equations xi = xi(t), i = 1, 2, ..., n for an arbitrary
parameter t, then the equations of a geodesics are given by
d2xi
dt2+Gi
(x,dx
dt
)= 0,
where
Gi =1
4gil[F 2]xkyl
yk −[F 2]xl
are called the spray coefficients of F n.
1.4 Connections and Covariant Differentiations
1.4.1 Finsler connection
A Finsler connection FΓ on a manifold M is a triad (F ijk, N
ik, C
ijk) of h-connection
F ijk, a non-linear connectionN i
k and a vertical connection C ijk such that h-covariant
derivative Kij|k and v-covariant derivative Ki
j|k of a Finsler tensor field Kij is given
by
Kij|k =
δKij
δxk+Km
j Fimk −Ki
mFmjk
11
and
Kij|k =
∂Kij
∂yk+Km
j Cimk −Ki
mCmjk,
whereδ
δxk≡ ∂
∂xk−N j
k
∂
∂yj.
The components of five torsion tensors T ijk, R
ijk, C
ijk, P
ijk, S
ijk and three curva-
ture tensors in terms of coefficients FΓ = (F ijk, N
ij , C
ijk) of the Finsler connection
can be written as:
Torsion Tensors
(h) h-torsion:
T ijk = F i
jk − F ikj, (1.1)
(v) h-torsion:
Rijk =
δN ij
δxk− δN i
k
δxj, (1.2)
(h) hv-torsion: Cijk,
(v) hv-torsion:
P ijk = ∂kN
ij − F i
kj, (1.3)
(v) v-torsion:
Sijk = C i
jk − C ikj. (1.4)
Curvature Tensors
h-curvature:
Rihjk =
δF ihj
δxk− δF i
hk
δxj+ Fm
hjFimk − Fm
hkFimj + Ci
hmRmjk, (1.5)
hv-curvature:
P ihjk = ∂kF
ihj − Ci
hk|j + CihmP
mjk , (1.6)
v-curvature:
Sihjk = ∂kC
ihj − ∂jC
ihk + Cm
hjCimk − Cm
hkCimj. (1.7)
The deflection tensor field Dik of a Finsler connection FΓ is given by
Dik = yjF i
jk −N ik. (1.8)
12
The Ricci identities w.r.to FΓ can be written as:
Khi|j|k −Kh
i|k|j = KriR
hrjk −Kh
rRrijk −Kh
i|rTrjk −Kh
i|rRrjk,
Khi|j|k −Kh
i|k|j = Kri P
hrjk −Kh
r Prijk −Kh
i|rCrjk −Kh
i|rPrjk,
Khi |j|k −Kh
i |k|j = Kri S
hrjk −Kh
r Srijk −Kh
i|rSrjk.
In Riemannian geometry, there is a unique connection, called the Riemannian
connection, which was introduced by Levi-Civita using Christoffel symbols. It has
two remarkable properties.
(1) The connection is compatible with the metric. In other words, the covariant
derivative gij|k of the metric tensor vanishes. This means
gij|k =∂gij(x)
∂xk− grj(x)Γ
rik(x)− gir(x)Γ
rjk(x) = 0.
Here Γijk denote the Christoffel symbols and given by
Γijk =
1
2gri(x)
(∂grj(x)
∂xk+∂grk(x)
∂xj− ∂gjk(x)
∂xr
).
(2) Torsion is zero. This means that Γijk(x) = Γi
kj(x).
Although in Finsler geometry there is no such connections that satisfy above two
conditions. However there are several connections in Finsler geometry such as
the Berwald connection, Cartan connection and Chern connection. In 1926, L.
Berwald was the first to introduce the concept of connection in Finsler geom-
etry. Berwald started his theory from the equations of geodesics to define the
Berwald connection. Berwald’s work stems from the study of systems of differen-
tial equations and is very much rooted in the calculus of variations. The Berwald
connection and Chern connection are not compatible with the metric, but it is
torsion free. The Cartan connection is compatible with the metric, but its torsion
is non-zero.
1.4.2 Berwald connection
Definition 1.3 The Berwald connection BΓ(Gijk, G
ij, 0) of a Finsler space
F n with Finsler metric F is a Finsler connection which is uniquely determined by
13
the following five axioms:
B1) (h) h−torsion free : T ijk = 0,
B2) F is h−metrical : F|i = 0,
B3) (h) hv−torsion free : C ijk = 0,
B4) (v) hv−torsion free : P ijk = 0,
B5) Deflection free : Dik = 0.
The connection coefficients of BΓ are given by
Gij = ∂jG
i, Gijk = ∂jG
ik,
where Gi are spray coefficients of the Finsler metric F .
The h-curvature tensor of Berwald connection BΓ is H ihjk, given by
H ihjk = π(jk)
[δkG
ihj +Gr
hjGirk
],
where π(jk) stands for interchange of indices j, k and subtraction.
The surviving curvature and torsion tensors of BΓ are as follows:
The hv-curvature tensor of Berwald connection BΓ is denoted by Bihjk also known
as Berwald curvature, given by
Bihjk = ∂hG
ijk.
The (h) torsion tensor Rijk is given by
Rijk = H i
0jk = δkGij − δjG
ik.
1.4.3 Cartan connection
It is an amusing irony that although Finsler geometry starts with only a norm
in any given tangent space, it regains an entire family of inner products, one for
each direction in that tangent space. This is why one can still make sence of
metric-compatibility in the Finsler setting.
Cartan (1934) introduced a connection called the Cartan connection which
was metric-compatible but not torsion free. The Cartan connections remain, to
this day, immensely popular with the Matsumoto and the Miron schools of Finsler
14
geometry. Besides the curvature tensors of hh-and hv- type, there is a third
curvature tensor associated with the Cartan connection. It is of vv- type. In
Cartan’s theory of Finsler spaces we have three curvature tensors Rihjk, P i
hjk, Sihjk
and three torsion tensors Rijk(= yhRi
hjk), P ijk(= yhP i
hjk), Cijk.
Definition 1.4 The Cartan connection CΓ of a Finsler space F n with Finsler
metric F is a Finsler connection which is uniquely determined by a system of five
axioms:
C1) h−metrical, i.e., gij|k = 0,
C2) (h) h−torsion free, i.e., T ijk = 0,
C3) Deflection free, i.e., Dik = 0,
C4) v−metrical, i.e., gij|k = 0,
C5) (v) v−torsion free, i.e., Sijk = 0.
The coefficients (Γijk, N
ij , C
ijk) of CΓ are as follows:
1. Γijk = girΓjrk,
where
Γijk =1
2(δkgij + δigjk − δjgik)
and
δk = ∂k −N rk ∂r.
2. N ij = γi0j − Ci
jrγr00,
where
γijk =1
2(∂kgij + ∂igjk − ∂jgik)
and the subscript 0 stands for the transvection by y.
3. C ijk = girCjrk,
where
Cijk =1
2∂kgij.
15
The surviving torsion tensors of CΓ are given by:
(v) h-torsion
Rijk = Ri
0jk,
(h) hv-torsion
Cijk =1
2∂kgij
and (v) hv-torsion
Pijk = Cijk|0.
The curvature tensors of CΓ are as follows:
h-curvature Rihjk satisfy
Rhijk = −Rihjk = −Rhikj,
hv-curvature
Phijk = π(hi)[Cjik|h + Cr
hjPrik
]and v-curvature
Shijk = π(jk) [CrhkCrij] .
1.4.4 Chern connection
S. S. Chern introduced a connection called the Chern connection, Chern (1943).
S. S. Chern studied the equivalence problem for Finsler spaces using the Cartan’s
exterior differentiation method. He discovered a very simple connection. Later on,
H. Rund independently introduced this connection in a different setting. Thus,
Chern’s connection was also called the Rund connection in literatures.
Definition 1.5 There is a unique linear connection, called Chern connection
on a Finsler manifold satisfying the following axioms:
1. Torsion freeness, i.e.,
Γijk = Γi
kj.
16
2. Almost metric compatibility, i.e.,
dgij = gikwkj + gkjw
ki + 2Cijkδy
k,
i.e., gij|k = 0 and gij;k = 2FCijk.
In this case, the connection coefficients Γijk are same as that of the Cartan con-
nection, i.e.,
Γijk =
gis
2
(δgsjδxk
− δgjkδxs
+δgksδxj
).
The surviving curvature tensors, h−curvature tensor Rihjk and hv−curvature ten-
sor P ihjk are given by
Rihjk =
δΓihk
δxj−δΓi
hj
δxk+ Γi
ljΓlhk − Γi
lkΓlhj
and
P ihjk = −F
∂Γihj
∂yk,
respectively.
The surviving torsion tensors are
Rijk = lhRi
hjk
and
P ijk = lhP i
hjk.
1.5 Flag curvature and S-curvature in Finsler ge-
ometry
1.5.1 Flag curvature
First of all in this section, we discuss flag curvature, for Finsler metrics. L. Berwald
first successfully extended the notion of Riemann curvature to Finsler spaces. The
flag curvature in Finsler geometry is an extension of the sectional curvature in
17
Riemannian geometry. For a Finsler space, at each point on the manifold the flag
curvature is a function of a tangent plane and a vector in the plane called the pole
vector.
We say that a Finsler metric is of scalar curvature if the flag curvature is
independent of the tangent planes containing the pole vectors. So the scalar
curvature is a function on the tangent bundle. In dimension two, every Finsler
metric is of scalar curvature.
If the flag curvature is constant then the Finsler metric is said to be of constant flag
curvature. One of the important problems in Finsler geometry is to characterize
Finsler metrics of scalar curvature.
Definition 1.6 For a Finsler metric F , the Riemann curvature Ry : TxM →
TxM is defined by Ry(u) = Rik(x, y)u
k ∂∂xi , u = uk ∂
∂xk , where
Rik = 2
∂Gi
∂xk− yj
∂2Gi
∂xj∂yk+ 2Gj ∂2Gi
∂yj∂yk− ∂Gi
∂yj∂Gj
∂yk.
Definition 1.7 For a tangent plane P ⊂ TxM containing y, the flag curva-
ture K(x, y, P ) with pole vector y is defined by
K(x, y, P ) :=gy(Ry(u), u)
gy(y, y)gy(u, u)− gy(y, u)gy(y, u),
where u ∈ P such that P = span y, u.
If K(x, y, P ) = K(x, y), then the Finsler metric is said to be of scalar flag curva-
ture.
If K(x, y, P ) = K(x), then the Finsler metric is said to be of isotropic flag curva-
ture.
If K(x, y, P ) = 3θF+ c(x), where c = c(x) is a scalar functions on M and θ is an
exact form on M , then the Finsler metric F is said to of almost isotropic flag
curvature.
If K(x, y, P ) = constant, then the Finsler metric is said to be of constant flag
curvature.
18
1.5.2 S-curvature
For a vector y ∈ TxM\ 0, let c = c(t) be the geodesic with c(0) = x and c(0) = y.
Then the S-curvature of the Finsler metric F is defined by
S(x, y) :=d
dt[τF (c(t), c(t))] |t=0,
where τF is called distortion of the Finsler metric F and defined by
τF = ln
√det(gij)
σF
and
σF =V ol(Bn)
V ol (yi) ∈ Rn|F (yibi) < 1.
The S-curvature of Finsler metric F is rewritten as
S =∂Gm
∂ym(x, y)− ym
∂ (logσF (x))
∂xm.
A Finsler metric F is said to have isotropic S-curvature if S = (n + 1)c(x)F , for
some scalar function c(x) on M . Further, if c is a constant, then F is said to be
of constant S-curvature.
F is said to be of almost isotropic S-curvature if
S = (n+ 1) cF + η ,
where η = ηi(x)yi is a 1-form and c = c(x) is a scalar function on M .
1.6 Some special Finsler spaces
1.6.1 Locally Minkowksi space
If a Finsler space has a covering of coordinate neighborhoods in which fundamental
metric tensor gij do not depend on xi, then it is called Locally Minkowksi.
Remark A Finsler space is a Locally Minkowksi space, iff one of the following
two conditions are satisfied:
(1) H ihjk = Gi
hjk = 0 in BΓ,
19
(2) Rihjk = Ci
hj|k = 0 in CΓ.
Thus, a Finsler space is locally Minkowski, iff it is a Berwald space and the h-
curvature tensor of BΓ or CΓ vanishes.
1.6.2 Berwald space
A Finsler space is called Berwald space if the spray coefficients Gi = 12Γijk(x)y
jyk
are quadratic in y, i.e., the coefficients of Berwald connection Gijk is independent
of y. Riemannian metrics are special Berwald metrics. It is well known that S-
curvature of a Berwald metric vanishes identically. Thus a Finsler metric with
vanishing S-curvature may be regarded as generalised Berwald metric.
1.6.3 Landsberg space
The Landsberg tensor is defined as
Ljkl = −1
2FFyi
∂3Gi
∂yj∂yk∂yl.
Finsler spaces with Ljkl = 0 are called Landsberg spaces. Obviously if the spray
coefficients Gi are quadratic in y for any x, then Ljkl = 0. Thus every Berwald
metric is a Landsberg metric . Landsberg metric can also be regarded as an
generalized Berwald metric. It is long open existing problem whether or not any
Landsberg metric is a Berwald metric.
1.6.4 Douglas space
In projective geometry of Finsler manifolds there is an important quantity called
Douglas tensor and defined by
Dihjk =
∂3πi
∂yh∂yj∂yk,
where
πi = Gi − 1
n+ 1
∂Gm
∂ymyi.
In local coordinate system the following three conditions are equivalent
Dihjk = 0.
20
Gi =1
2Γijk(x)y
jyk + P (x, y)yi.
Dij = Giyj −Gjyi = Aijklm(x)y
kylym.
A Finsler metric is called a Douglas metric if Dihjk = 0. Douglas metric is another
generalization of Berwald metric.
1.6.5 Weakly-Berwald space
The E-curvature is defined by the trace of the Berwald curvature, i.e., Eij =12Bm
mij.
A Finsler space is said to be Weakly-Berwald space if, Eij =12Bm
mij = 0. Weakly-
Berwald space with an (α, β)-metric has been investigated by several authors [Lee
and Lee (2006) and Matsumoto (1991)].
1.7 KCC theory
The notion of the KCC (Kosambi-Cartan-Chern) theory was initiated by Kosambi
(1933), Cartan (1933) and Chern (1939), and the abbreviation KCC comes natu-
rally from the names of these three initiators, who profound the geometric theory of
a system of second order ordinary differential equation (SODE). The first attempt
to establish and to develop systematically the KCC theory is due to Antonelli and
Bucataru (2001). The KCC theory describes the evolution of a dynamical system
in geometric terms, by considering it as a geodesic in a Finsler space.
The most significant applications of the KCC theory have been developed for
second-order autonomous systems. For several such systems which provide Lotka-
Volterra models from biology, the Jacobi stability has been investigated. Recent
advances have been obtained in the Riemannian KCC frame-work.
1.7.1 Linear stability vs Jacobi stability
A second order differential equation can be investigated in geometric terms by
KCC-theory inspired by the geometry of a Finsler space. By associating a non-
linear connection and a Berwald type connection to the differential system, five
21
geometrical invariants are obtained with the second invariant giving the Jacobi
stability of the system.
The Jacobi stability gives a simple and powerful tool for constraining the
physical properties of different systems such as dynamical systems, which is given
by second order differential equations (SODE) [Bohmer et al. (2010)].
The study of Jacobi stability is complementary to the study of linear stability
(Lyapunov’s) and is based on the study of Lyapunov stability of whole trajectories
in a region, and hence the perturbation yields trajectories close to the reference
trajectory.
Similarly, in the case of Lyapunov stability, the perturbations of a stable
equilibrium point lead to trajectories which will be dumped out, provided that
these are small enough so as not to escape from a basin of attraction [Antonelli et
al. (1993)].
Linear stability analysis involves the linearisation of the dynamical system
via the Jacobian matrix of a non-linear system, while the KCC theory addresses
stability of a whole trajectory in a tubular region given by Bohmer et al. (2010).
The Jacobi stability gives a global stability than the linear stability. The
Jacobian matrix of the linearised system plays an important role in linear stability
analysis. In linear stability we consider the signs of the eigen values of Jacobian
matrix of corresponding linear system at equilibrium point, where as in Jacobi
stability we consider the signs of the eigenvalues of the deviation curvature tensor
P ij evaluated at the same point.
1.8 Mathematical Cosmology in Finslerian space-
time
One of the main aims of Cosmology is to express all known forces of nature in
one unified theory. Practically, all unification efforts nowadays proceed from the
assumption that quantum field theory is fundamental and gravitation must be
squeezed into a quantum context.
22
On the other hand, there exist other approaches to unification which uses
some geometrical theories. They assumed that a geometrical theory, Einstein’s
general relativity is a fundamental theory, thus electromagnetism and other fields
can be unified by means of a geometrical theory.
The most known geometrical approaches to unification are the theories of
Weyl and Kaluza-Klein, which aim to geometrize electromagnetism like gravita-
tion. These theories faced with serious problems such as, in Weyl’s theory, the
norms of vectors are not invariant under parallel transport, and in the approach
of Kaluza-Klein theories, electrodynamics is geometrized in a five dimensional
space-time. Also quantization of space-time is another existing problem when the
electromagnetic field is quantized.
Finsler geometry is the geometry of space and motion. In our universe we remark
that “there is no position without motion ”. A Finsler space can be considered as
a manifold of positions (coordinate systems xi) and of tangent vectors yi (veloci-
ties) along the curves (world lines of the moving particles) of the background. The
general spaces of paths are closely connected with the principle of equivalence.
In the four dimensional world of space-time the trajectory of a particle falling
freely in a gravitational field is a certain fixed curve. Its direction at any point
depends on the velocity of the particle. The principle of equivalence implies that
there is a preferred set of curves in space-time at any point, pick up any direction
and there is a unique curve in that direction that will be trajectory of any particle
starting with that velocity. These trajectories are thus the properties of space-time
itself.
This standpoint reveals a profound relation between the principle of equiva-
lence and the space-time of paths in Finsler spaces. In addition with a Finsler-
Randers type space-time, as we shall present in the following, the limits of the
equivalence principle of General Relativity can be extended since the presence of
the electromagnetic field does not affect the geodesic motion of a charged par-
ticle in the space. The electromagnetic field is intrinsically incorporated in the
geometry of the space.
23
Chapter 2On Finsler space with a special (α, β)-metric
The purpose of the present chapter is to study a special (α, β)-metric, which is
considered as a generalization of the Randers metric as well as of the Z. Shen’s
square metric and to find the conditions for a Finsler space with this special metric
to be a Berwald space, a Douglas space and Weakly-Berwald space respectively.
2.1 Introduction
The notion of (α, β)-metric in Finsler spaces was introduced by Matsumoto (1972)
as a generalization of Randers metric F = α + β, where α is a regular Rieman-
nian metric α =√aij(x)yiyj, i.e., det(aij) = 0 and β is a one-form β = bi(x)y
i
and studied by many authors [Matsumoto (1991)and Matsumoto (1992)]. Some
authors also assume α to be positive definite [Shen and Yildirim (2008) and Lee
and Lee (2006)].
A Finsler metric F on a differentiable manifold M is called an (α, β)-metric,
if F is a positively homogeneous function of degree one in α and β. Other than
Randers metric F = α + β, there are several important (α, β)-metrics, namely
Kropina metric F = α2
β, Matsumoto metric F = α2
α−β, generalized Kropina metric
F = αn+1
βn and Z. Shen’s square metric F = (α+β)2
α. Z. Shen’s square metric is also
interesting because the metric,
F (x, y) =
(√(1− |x|2)|y|2 + ⟨x, y⟩2 + ⟨x, y⟩
)2(1− |x|2)2
√(1− |x|2)|y|2 + ⟨x, y⟩2
, (x, y) ∈ TRn
24
constructed by L. Berwald in 1929, which is projectively flat on unit ball Bn with
constant flag curvature K = 0, can be written in form F = (α+β)2
αfor some suitable
α and β. Here | · | and ⟨, ⟩ denote the standard Euclidean norm and inner product
respectively on Rn. Shen and Yildirim (2008) also introduced a special (α, β)-
metric F = α+ ϵβ+ k β2
α, which may be considered as a generalization of Randers
metric (k = 0, ϵ = 1) and square (α, β)-metric (k = 1, ϵ = 2). If the Riemannian
metric α is positive definite then in view of Chern and Shen (2004), F is positive
definite with ||β|| < b0 iff 1+ ϵs+ks2 > 0 and 1+2kb2−3ks2 > 0 for any numbers
s and b with |s| ≤ b < b0 where s = βα. In particular Z. Shen square metric is a
positive definite Finsler metric if ||β||α < 1. Z. Shen and Yildirim classified this
special (α, β)-metric under the restriction of projectively flatness and constant flag
curvature. Recently Zhou (2010) proved that a square metric with constant flag
curvature must be projectively flat and hence classified all projectively flat square
(α, β)-metrics. For the rest of the chapter we assume k = 0, ϵ = 0.
Definition 2.1 A Finsler space is called Berwald space if the spray coeffi-
cients Gi = 12Γijk(x)y
jyk are quadratic in y, i.e., the coefficients of Berwald con-
nection Gijk is independent of y.
Riemannian metrics are special Berwald metrics. Weakly-Berwald space and Dou-
glas spaces are generalizations of Berwald spaces.
Definition 2.2 The tensor D := Dijkl∂i ⊗ dxj ⊗ dxk ⊗ dxl is called Douglas
curvature, introduced by Douglas (1927-28), where
Dijkl :=
∂3
∂yj∂yk∂yl
(Gi − 1
n+ 1
∂Gm
∂ymyi).
Douglas curvature, always vanishes for a Riemannian metrics. Finsler metrics
with vanishing Douglas curvature are called Douglas metric and the space is called
Douglas space.
Definition 2.3 The E-curvature is defined by the trace of the Berwald cur-
vature, i.e., Eij =12Bm
mij. A Finsler space is said to be Weakly-Berwald space if,
Eij =12Bm
mij = 0.
25
Weakly-Berwald space with an (α, β)-metric has been investigated by several au-
thors [Lee and Lee (2006) and Matsumoto (1991)]. A Finsler space with an (α, β)-
metric is a Weakly-Berwald space iff Bmm = ∂Bm
∂ymis a one form, i.e., Bm
m is a homo-
geneous polynomial in (yi) of degree one. Further Matsumoto (1991) investigated
that a Finsler space with an (α, β)-metric is a Weakly-Berwald space, iff Bm are
homogeneous polynomials in (yi) of degree 2.
The purpose of the present chapter is to find the conditions for a Finsler space
with special (α, β)-metric F = α + ϵβ + k β2
αto be Berwald space, Douglas space
and Weakly-Berwald space.
Let M be an n-dimensional C∞-manifold. TxM denotes the tangent space of M
at x. The tangent bundle of M is the union of tangent spaces TM :=∪
x∈M TxM .
We denote the elements of TM by (x, y) where y ∈ TxM . Let TM0 = TM \ 0.
Definition 2.4 A Finsler metric on M is a function F : TM → [0,∞) with
the following properties:
(i) F is C∞ on TM0,
(ii) F is positively 1-homogeneous on the fibers of tangent bundle TM ,
and (iii) the Hessian of F 2
2with element gij(x, y) = 1
2∂2F 2
∂yi∂yjis regular on TM0,
i.e., det(gij) = 0.
The pair (M,F ) is then called a Finsler space. F is called fundamental function
and gij is called fundamental tensor. Let Cijk =12
∂gij∂yk
be Cartan tensor. Consider
the Finsler space F n = (M,F ) equipped with an (α, β)-metric F (α, β). Let γijkdenote the Christoffel symbols in the Riemannian space (M,α). Denote by bi;j, the
covariant derivative of the vector field bi with respect to Riemannian connection
γijk, i.e., bi;j = ∂bi∂xj − bkγ
ijk.
Consider the following notations from Chern and Shen (2004)
rij =1
2bi;j + bj;i , rij = aihrhj, rj = bir
ij,
sij =1
2bi;j − bj;i , sij = aihshj, sj = bis
ij,
bi = aihbh, b2 = bibi.
26
2.2 The condition to be a Berwald space
In the present section, we find the condition that a Finsler space F n with a special
(α, β)-metric
F = α + ϵβ + kβ2
α(2.1)
where ϵ, k are non zero constants, to be a Berwald space.
A Finsler space is called Berwald space if the coefficients Gij of Berwald con-
nection BΓ are linear. If the spray coefficients Gi of a Finsler space with an
(α, β)-metric are given by 2Gi = γi00 + 2Bi, then we have Gij = γi0j + Bi
j and
Gijk = γijk + Bi
jk, where Bijk = ∂kB
ij and Bi
j = ∂jBi. Thus a Finsler space with
an (α, β)-metric is a Berwald space iff Gijk = Gi
jk(x), equivalently Bijk = Bi
jk(x).
Moreover on account of Matsumoto (1991) Bijk is determined by
FαBtjiy
jyt+αFβ(Btjibt− bj;i)y
j = 0 where yk = aikyi. (2.2)
For the special (α, β)-metric (2.1) we have,
Fα = 1− kβ2
α2, Fβ = ϵ+
2kβ
α, Fαα =
2kβ2
α3, Fββ =
2k
α. (2.3)
Substituting (2.3) in (2.2) equation, we have
(α2 − kβ2)Btjiy
jyt + α2(αϵ+ 2kβ)(Btjibt − bj;i)y
j = 0. (2.4)
Assume that F n is a Berwald space, i.e., Bijk = Bi
jk(x).
Separating (2.4) in rational and irrational terms of yi as
(α2 − kβ2)Btjiy
jyt + 2kα2β(Btjibt − bj;i)y
j + α3ϵ(Btjibt − bj;i)y
j = 0 (2.5)
which yields two equations
(α2−kβ2)Btjiy
jyt+2kα2β(Btjibt−bj;i)yj
(2.6)
and
(Btjibt − bj;i)y
j = 0, α = 0, ϵ = 0. (2.7)
27
Substituting (2.7) in (2.6), we have
(α2 − kβ2)Btjiy
jyt = 0. (2.8)
Case (i) If Btjiy
jyt = 0, we have
Btjiath +Bt
hiatj = 0 and Btjibt − bj;i = 0. (2.9)
Thus we obtain Btji = 0 by Christoffel process in the first equation of (2.9) and
from second of (2.9), we have bi;j = 0.
Case (ii) If (α2 − kβ2) = 0
⇒ α is a one form, which is a contradiction.
Conversely, if bi;j = 0, then Btji = 0 are uniquely determined from (2.4).
Hence, we conclude the following
Theorem 2.1 A Finsler space with a special (α, β)-metric F = α+ ϵβ+k β2
α
where ϵ, k are non zero constants, is a Berwald space iff bi;j = 0.
2.3 The condition to be a Douglas space
In this section, we find the condition for a Finsler space F n with a special (α, β)-
metric (2.1), to be of Douglas type.
Bacso and Matsumoto (1997) characterizes a Douglas space as a Finsler space for
which Bij = Biyj − Bjyi are homogeneous polynomials of degree 3, in short we
write Bij is hp(3).
In view of Matsumoto (1991), if β2Fα +αγ2Fαα = 0, then the function Gi(x, y) of
F n with an (α, β) -metric is written in the form
2Gi = γi00 + 2Bi,
where
Bi =αFβs
i0
Fα
+ C∗βFβy
i
αF− αFαα
Fα
(yi
α− αbi
β
),
28
C∗ =αβ(r00Fα − 2s0αFβ)
2(β2Fα + αγ2Lαα),
and
γ2 = b2α2 − β2.
The vector Bi(x, y) is called the difference vector. Hence Bij is written as
Bij =αFβ
Fα
(si0yj − sj0y
i) +α2Fαα
βFαC∗(biyj − bjyi).
Substituting (2.3) in above equation, we get
[α2(1 + 2kb2)− 3kβ2][(α2 − kβ2)Bij − α2(ϵα+ 2kβ)(si0yj − sj0y
i)]
−kα2[r00(α2 − kβ2)− 2s0α
2(ϵα + 2kβ)](biyj − bjyi) = 0. (2.10)
If F n is a Douglas space, that is, Bij are hp(3). Arranging the rational and
irrational terms, equation (2.10) can be re written as
[(α2 − kβ2)Bij − 2kβα2(si0yj − sj0y
i)][α2(1 + 2kb2)− 3kβ2]
−kα2[r00(α2 − kβ2)− 4ks0α
2β)](biyj − bjyi)
−α[α2(1 + 2kb2)− 3kβ2
α2ϵ(si0y
j − sj0yi)− 2s0kα
4ϵ(biyj − bjyi)] = 0. (2.11)
Separating rational and irrational terms of yi in (2.11) we have the following two
equations
[(α2 − kβ2)Bij − 2kβα2(si0yj − sj0y
i)][α2(1 + 2kb2)− 3kβ2]
−kα2[r00(α2 − kβ2)− 4ks0α
2β)](biyj − bjyi) = 0 (2.12)
and
α2(1 + 2kb2)− 3kβ2
ϵ(si0y
j − sj0yi)
−2s0kα2ϵ(biyj − bjyi) = 0. (2.13)
Substituting (2.13) in (2.12), we have
[α2(1 + 2kb2)− 3kβ2](α2 − kβ2)Bij
+α2r00(k2β2 − kα2)(biyj − bjyi) = 0. (2.14)
29
Only the term 3k2β4Bij of (2.14) does not contain α2.
Hence we must have hp(5), vij5 satisfying
3k2β4Bij = α2vij5 . (2.15)
Now, we study the following two cases:
Case (i) α2 0(modβ).
In this case, (2.15) is reduced to Bij = α2vij, where vij are hp(1).
Thus (2.14) gives
[α2(1 + 2kb2)− 3kβ2]vij − kr00(biyj − bjyi) = 0. (2.16)
Transvecting this by biyj, where yj = ajkyk, we have
α2[(1 + 2kb2)vijbiyj − kb2r00] = β2(−kr00 + 3kvijbiyj). (2.17)
Since α2 0(modβ), there exists a function h(x) satisfying
[(1 + 2kb2)vijbiyj − kb2r00] = h(x)β2,
(−kr00 + 3kvijbiyj) = h(x)α2.
Eliminating vijbiyj from the above two equations, we obtain
r00(−k + k2b2) = h(x)[(1 + 2kb2)α2 − 3kβ2]. (2.18)
From (2.18), we get
bi;j = l [(1 + 2kb2)aij − 3kbibj], (2.19)
where l = h(x)k(1−kb2)
provided b2 = 1k. Hence bi is a gradient vector.
Conversely, if (2.19) holds, then sij = 12(bi;j − bj;i) = 0 and we get (2.18).
Therefore, (2.12) is written as follows
Bij = −lk[α2(biyj − bjyi)]
which are hp(3), that is, F n is a Douglas space.
Case (ii) α2 ∼= 0(modβ).
30
Consider the following Lemma.
Lemma. [Lee and Park (2004)] If α2 ∼= 0(modβ), that is, aij(x)yiyj contains
biyi as a factor, then the dimension n is equal to 2 and b2 vanishes. In this case
we have 1-form δ = di(x)yi satisfying α2 = βδ and dib
i = 2. In this case the
equation (2.15) is reduced to Bij = δwij2 , where wij
2 are hp(2).
Hence the equation (2.13) leads to
−2s0kδϵ(biyj − bjyi) + ϵ[δ(−3kβ](si0y
j − sj0yi) = 0. (2.20)
Transvecting the above equation by yibj, we get −s0ϵβδ[−kβ + δ] = 0 but β and
δ are non zero and β = δk, we have ⇒ s0 = 0. Substituting s0 = 0 in equation
(2.20), we have sij = 0. Therefore, (2.16) reduces to
(δ − 3kβ)wij2 − kr00(b
iyj − bjyi) = 0.
Transvecting the above equation by biyj, we get (δ − 3kβ)wij2 biyj + kr00β
2 = 0,
which can be written as δwij2 biyj = −kβ(βr00 − 3wij
2 biyj). Therefore, there exists
an hp(2), λ = λijyiyj such that wij
2 biyj = −kβλ, βr00 − 3wij2 biyj = δλ.
Eliminating wij2 biyj from these equations, we get
βr00 = λ(δ − 3kβ), (2.21)
which implies there exists an hp(1), v0 = vi(x)yi such that
r00 = v0(δ − 3kβ), λ = v0β. (2.22)
In view of equation (2.22) and considering sij = 0, we have
bi;j = rij =1
2[vi(dj − 3kbj) + vj(di − 3kbi)]. (2.23)
Hence bi is a gradient vector.
Conversely, if (2.23) holds, then sij = 0, which implies r00 = v0(δ − 3kβ).
Therefore, (2.10) is written as Bij = v0kδ(biyj − bjyi), which are hp(3), i.e., F n is
a Douglas space.
Thus, we have
31
Theorem 2.2 A Finsler space with a special (α, β)-metric F = α+ ϵβ+k β2
α
where ϵ, k are non zero constants, is a Douglas space iff either
(1) α2 0(modβ), b2 = 1k: bi;j is writtten in the form (2.19),
or (2) α2 ∼= 0(modβ) : n = 2 and bi;j is writtten in the form (2.23),
where α2 = βδ, δ = di(x)yi, v0 = vi(x)y
i.
Remark: If α is a positive definite Riemannian metric then the set of non Rie-
mannian (α, β)-metric satisfying case (ii), i.e., α2 ∼= 0(modβ) is void, otherwise
b2 = 0 ⇒ bi = 0 that is β = 0. For positive definite case Li et al. (2009) have
found a condition for a general (α, β)- metric to be Douglas. Comparing the re-
sults of Theorem (1.1) [Li et al. (2009)], we have k1 = 2k, k2 = 0, k3 = −3k
and bi;j = 2τ (1 + 2kb2)aij − 3kbibj. Thus we have τ = l(x)2
, where l is given by
l = h(x)k(1−kb2)
provided b2 = 1k.
2.4 The condition to be a Weakly-Berwald space
In the present section, we consider the condition that the Finsler space with an
(α, β)- metric (2.1), to be a Weakly-Berwald space. Weakly-Berwald space is a
generalization of Berwald space, introduced by M. Matsumoto and studied by
several authors [Lee and Lee (2006) and Matsumoto (1992)].
The spray coefficients Gi(x, y) of F n with an (α, β)-metric, given by Lee and Lee
(2006), can be written as
2Gm = γm00 + 2Bm,
where
Bm =
(E∗
α
)ym +
(αFβ
Fα
)sm0 −
(αFαα
Fα
)C∗(
ym
α
)−(α
β
)bm, (2.24)
where
E∗ =
(βFβ
F
)C∗, C∗ =
αβ (r00Fα − 2αs0Fβ)2 (β2Fα + αγ2Fαα)
, γ2 = b2α2 − β2. (2.25)
Differentiating (2.24) w.r.t yn and summing over suffixes m and n, we have
Bmm =
1
2αF (βFα)2Ω
22Ω2AC∗ + 2αFΩ2Bs0 + α2FFαFαα (Cr00 +Ds0 + Er0)
,
(2.26)
32
where
A = (n+ 1) β2Fα (βFαFβ − αFFαα) + αγ2Fα(Fαα)
2 − 2FαFαα − αFαFααα
,
B = α2FFαα,
C = βγ2−β2(Fα)
2 + 2b2α3FαFαα − α2γ2(Fαα)2 + α2γ2FαFααα
,
(2.27)
D = 2αβ3(γ2 − β2
)FαFβ − α2β2γ2FαFαα
−2αβγ2(γ2 + 2β2
)FβFαα − α3γ4(Fαα)
2 − α2βγ4FβFααα
,
E = 2α2β2FαΩ
and
Ω =(β2Fα + αγ2Fαα
),
(2.28)
provided Ω = 0.
Substituting (2.3) in equation (2.24), (2.25), (2.27) and (2.28), we have
Bm = C∗[
β (αϵ+ 2kβ)
α(α2 + ϵαβ + kβ2)− 2kβ2
α (α2 − kβ2)
ym +
2kβα
(α2 − kβ2)
bm]
+
α2 (αϵ+ 2kβ)
(α2 − kβ2)
sm0 , (2.29)
where
C∗ =αβ r00 (α2 − kβ2)− 2α2s0 (αϵ+ 2kβ)2 β2 (α2 − kβ2) + 2kβ2 (b2α2 − β2)
.
From (2.3) and (2.27), we get
A =(n+ 1) β3 (α2 − kβ2)
α5
[α3ϵ− 3kϵαβ2 − 4k2β3
]+
2γ2kβ2
α5
×(α2 + ϵαβ + kβ2
) (α2 + kβ2
),
B =(α2 + ϵαβ + kβ2) (2kβ2)
α2,
C =β3γ2
α4
[−(α2 − kβ2
)2+ 4b2kα2
(α2 − kβ2
)− 2γ2
(3kα2 − k2β2
)],
(2.30)
33
D =2β3
α2
[(γ2 − β2
) (α2 − kβ2
)(αϵ+ 2kβ)− 2k
(α2 − kβ2
)βγ2]
−[4kγ2
(γ2 + 2β2
)(αϵ+ 2kβ)− 2γ4
(4k2β + 3kϵα
)],
Ω =β2
α2
[α2(1 + 2kb2
)− 3kβ2
],
E =2β4
α2
(α2 − kβ2
) [(1 + 2kb2
)α2 − 3kβ2
]and
C∗ =α r00 (α2 − kβ2)− 2α2s0 (αϵ+ 2kβ)
2β α2 (1 + 2kb2)− 3kβ2.
Substituting (2.30) into (2.26), we getα10β
(8b4k2 + 8b2k + 2
)+ α4β7
(8b4k5 + 32b2k4 − 4k3
)+ α6β5
(−8b4k4 + 16b2k3 + 28k2
)+ α8β3
(−8b4k3 − 32b2k2 − 14k
)+ 18k5β11 + α2β9
(−24b2k5 − 30k4
)+ α9β2
(8b4k2ϵ+ 8b2kϵ+ 2ϵ
)+ α7β4
(−16b4k3ϵ− 40b2k2ϵ− 16kϵ
)+ α5β6
(8b4k4ϵ+ 56b2k3ϵ+ 44k2ϵ
)+ αβ10
(18k4ϵ
)+ α3β8
(−48k3ϵ− 24b2k4ϵ
)Bm
m
+α6β4
(−20b2k3 − 16k2 − 8b2k3n− 4k2n
)+ α4β6
(4b2k4 + 32k3 + 16b2k4n+ 20k3n
)+ α8β2
(12k2b2 + 8b4k3
)+ α2β8
(−16k4 − 8b2k5n− 28k4n+ 12b2k5
)+ β10
(16k5 + 12k5n
)+α7β3
(2k2b2ϵ− 8kϵ− 8knϵ− 10b2k2nϵ
)+ α9β
(2b2kϵ+ 2b2knϵ+ nϵ+ ϵ
)+ α5β5
(−10b2k3ϵ+ 10k2ϵ+ 14b2k3nϵ+ 22k2nϵ
)+ αβ9
(−3k4ϵ+ 9k4nϵ
)+ α3β7
(6b2k4ϵ− 6b2k4nϵ− 24k3nϵ
)r00
+α10β
(−4b2ϵ2k − 4b2ϵ2kn− 2nϵ2 − 2ϵ2 + 8b2k2 + 4k
)+α8β3
(−16k2 + 14kϵ2 + 14knϵ2 − 8b2k2ϵ2 + 16b2k2ϵ2n− 56k3b2
)+α6β5
(24b2k4 + 72k3 + 32b2k4n+ 16k3n+ 20k3b2ϵ2 − 6k2ϵ2 − 12b2k3nϵ− 30k2ϵ2n
)+α2β9
(−12k5 + 48k5n
)+ α4β7
(24b2k5 − 48k4 − 32b2k5n− 64k4n− 6k3ϵ2 + 18k3ϵ2n
)+α7β4
(−16b2k3ϵ+ 48b2k3nϵ+ 36k2nϵ+ 40k2ϵ
)+α5β6
(−16k3ϵ− 40b2k4nϵ− 92k3nϵ+ 40b2k4ϵ
)+ α9β2
(−32k2b2ϵ− 8b2k2nϵ− 4knϵ
)+α3β8
(−16k4ϵ+ 60k4nϵ
)s0 +
α10β
(8b2k2 + 4k
)+ α8β3
(−8b2k3 − 16k2
)+α6β5
(−8b2k4 + 8k3
)+ α2β9
(−12k5
)+ α4β7
(8b2k5 + 16k4
)+ α9β2
(8b2k2ϵ+ 4kϵ
)+α5β6
(8b2k4ϵ+ 28k3ϵ
)+ α3β8
(−12k4ϵ
)+ α7β4
(−16b2k3ϵ− 20k2ϵ
)r0 = 0.
(2.31)
34
Suppose that F n is Weakly-Berwald space, i.e., Bmm is hp(1). Since α is irrational
in (yi), the equation (2.31) is divided into two equations as follows
F1Bmm + βG1r00 + α2H1s0 + α2I1r0 = 0 (2.32)
βF2Bmm +G2r00 + α2βH2s0 + α2βI2r0 = 0, (2.33)
where
F1 =α10(8b4k2 + 8b2k + 2
)+ α4β6
(8b4k5 + 32b2k4 − 4k3
)+ α6β4
(−8b4k4 + 16b2k3 + 28k2
)+ α8β2
(−8b4k3 − 32b2k2 − 14k
)+ 18k5β10 + α2β8
(−24b2k5 − 30k4
),
F2 =α8(8b4k2ϵ+ 8b2kϵ+ 2ϵ
)+ α6β2
(−16b4k3ϵ− 40b2k2ϵ− 16kϵ
)+ α4β4
(8b4k4ϵ+ 56b2k3ϵ+ 44k2ϵ
)+ 18k4ϵβ8 + α2β6
(−48k3ϵ− 24b2k4ϵ
),
G1 =α6β2(−20b2k3 − 8b2k3n− 16k2 − 4k2n
)+ α4β4
(4b2k4 + 32k3 + 16b2k4n+ 20k3n
)+ α8
(12k2b2 + 8b4k3
)+ α2β6
(−16k4 − 8b2k5n− 28b4n+ 12b2k5
)+ β8
(16k5 + 12k5n
),
G2 =α8(2b2kϵ+ 2b2kϵn+ nϵ+ ϵ
)+ α6β2
(2k2b2ϵ− 8kϵ− 8knϵ− 10k2b2nϵ
)+ α4β4
(−10b2k3ϵ+ 10k2ϵ+ 14b2k3nϵ+ 22k2nϵ
)+ β8
(−3k4ϵ+ 9k4nϵ
)+ α2β6
(6b2k4ϵ− 6b2k4nϵ− 24k3nϵ
),
H1 =α8(−4b2ϵ2k − 4b2ϵ2kn− 2nϵ2 − 2ϵ2 + 8b2k2 + 4k
)+ α6β2
(−16k2 + 14kϵ2 + 14knϵ2 − 8b2k2ϵ2 + 16b2k2ϵ2n− 56k3b2
)+ α4β4
(24b2k4 + 72k3 + 32b2k4n+ 16k3n+ 20k3b2ϵ2 − 6k2ϵ2 − 12b2k3nϵ− 30k2ϵ2n
)+ β8
(−12k5 + 48k5n
)+ α2β6
(24b2k5 − 48k4 − 32b2k5n− 64k4n− 6k3ϵ2 + 18k3ϵ2n
),
H2 =α4β2(−16b2k3ϵ+ 48b2k3nϵ+ 36k2nϵ+ 40k2ϵ
)+ α2β4
(−16k3ϵ− 40b2k4nϵ− 92k3nϵ+ 40b2k4ϵ
)+ α6
(−32k2b2ϵ− 8k2b2nϵ− 4knϵ
)+ β6
(−16k4ϵ+ 60k4nϵ
),
I1 =α8(8b2k2 + 4k
)+ α6β2
(−8b2k3 − 16k2
)+ α4β4
(−8b2k4 + 8k3
)+ β8
(−12k5
)+ α2β6
(8b2k5 + 16k4
)35
and
I2 =α6(8b2k2ϵ+ 4kϵ
)+ α2β4
(8b2k4ϵ+ 28k3ϵ
)+ β6
(−12k4ϵ
)+ α4β2
(−16b2k3ϵ− 20k2ϵ
).
Eliminating Bmm from these equations, we obtain
Rr00 + α2βSs0 + α2βTr0 = 0, (2.34)
where
R = F1G2 − β2G1F2, S = F1H2 −H1F2, T = F1I2 − F2I1.
Since only the term (8b4k2 + 8b2k + 2) (2b2kϵ+ 2b2knϵ+ nϵ+ ϵ)α18r00 of Rr00 in
(2.34) does not contain β, we must have, hp(19)V19 such that
α18r00 = βV19. (2.35)
First we are concerned with α2 0(modβ) and b2 = 0. Hence (2.35) shows the
existence of a function V 1 satisfying V19 = V 1α18, and we get r00 = V 1β.
Then (2.34) is reduced to
RV 1 + α2Ss0 + α2Tr0 = 0. (2.36)
Only the term 18k5 (−3k4ϵ+ 9k4nϵ)− 18k4ϵ (16k5 + 12k5n) β18V 1 of the above
does not contain α2, and hence we must have hp(17), V17 satisfying
18k5
(−3k4ϵ+ 9k4nϵ
)− 18k4ϵ
(16k5 + 12k5n
)β18V 1 = α2V17.
Since α2 0(modβ), we have V17 = 0, i.e., V 1 = 0. Hence we obtain r00 =
0; rij = 0 and r0 = 0; rj = 0. Substituting V 1 = 0, r0 = 0 in (2.36), we get
Ss0 = 0 ⇒ s0 = 0 since S = 0.
Conversely, substituting r00 = 0, s0 = 0 and r0 = 0 into (2.31), we have Bmm = 0.
That is, the Finsler space with (2.1) is a Weakly-Berwald space. On the other
hand, we suppose that the Finsler space with (2.1) be a Berwald space. Then we
have r00 = 0, s0 = 0 and r0 = 0, because the space is a Weakly-Berwald space
from the above discussion. Substituting the above into (2.29), we have Bm = 0
that is, the Finsler space with (2.1) is a Berwald space. Hence sij = 0 hold good.
36
Theorem 2.3 A Finsler space with the metric (2.1) is Weakly-Berwald space
iff rij = 0 and sj = 0.
Remark: In view of Theorem (2.1) a Finsler space with metric (2.1) is a Berwald
space iff rij = 0 and sij = 0.
37
Chapter 3On Randers change of a Finsler spacewith m-th root metric
The purpose of the present chapter is to find a condition under which a Finsler
space with Randers change of m-th root metric is projectively related to a m-th
root metric and also to find a condition under which this Randers transformed m-
th root Finsler metric is locally dually flat. Moreover, if the transformed Finsler
metric is conformal to the m-th root Finsler metric, then it is proved that both of
them reduce to Riemannian metrics.
3.1 Introduction
The concept of m-th root metric was introduced by Shimada (1979), applied to
ecology by Antonelli et al. (1993) and studied by several authors [Srivastava and
Arora (2012), Tayebi et al. (2012), Tayebi and Najafi (2011), Yu and You (2010)
and Prasad and Dwivedi (2002)]. It is regarded as a generalization of Riemannian
metric in the sense that the second root metric is a Riemannian metric. For m = 3,
it is called a cubic Finsler metric studied by Matsumoto and Numata (1982) and
for m = 4, it is quartic metric studied by Li and Shen (2012). In four-dimension,
the special fourth root metric in the form F = 4√y1y2y3y4 is called the Berwald-
Moor metric. This metric has been studied by Balan (2010) and considered by
physicists as an important subject for a possible model of space time. Recent
studies show that m-th root Finsler metrics play a very important role in physics,
space-time and general relativity as well as in unified gauge field theory [Pavlov
(2006), Lebedev (2006), Balan and Brinzei (2006), Balan and Brinzei (2005) and
38
Asanov (1984)]. Li and Shen (2012) have studied the geometric properties of
locally projectively flat fourth root metrics in the form F = 4√aijkl(x)yiyjykyl and
generalized fourth root metrics in the form F =√√
aijkl(x)yiyjykyl + bij(x)yiyj.
Tayebi and Najafi (2011) have characterized locally dually flat and Antonelli m-
th root metrics and Tayebi et al. (2012) have found a condition under which a
generalized m-th root metric is projectively related to m-th root metric. Brinzei
(2009) has investigated necessary and sufficient condition for a Finsler space with
m-th root metric to be projectively related.
Recently, Srivastava and Arora (2012) have introduced Randers change of m-
th root metric and studied relations between various tensors of the transformed
Finsler space and Finsler space with m-th root metric. In this chapter a condition
under which the transformed Finsler space is projectively related with given Finsler
space has been investigated. Also the condition under which the transformed
Finsler space is locally dually flat has been found. Further it is proved that if the
transformed Finsler space and original Finsler space with m-th root metric are
conformally related then both reduce to Riemannian metrics.
3.2 Preliminaries
Let M be an n-dimensional C∞-manifold, TxM denotes the tangent space of M
at x. The tangent bundle TM is the union of tangent spaces, TM :=∪
x∈M TxM .
We denote the elements of TM by (x, y), where x = (xi) is a point of M and
y ∈ TxM called supporting element. We denote TM0 = TM \ 0.
Definition 3.1 A Finsler metric on M is a function F : TM → [0,∞) with
the following properties:
(i) F is C∞ on TM0,
(ii) F is positively 1-homogeneous on the fibers of tangent bundle TM and
(iii) the Hessian of F 2
2with element gij = 1
2∂2F 2
∂yi∂yjis positive definite on TM0.
The pair (M,F ) = F n is called a Finsler space. F is called the fundamental
function and gij is called the fundamental tensor of the Finsler space F n.
39
The normalized supporting element li, angular metric tensor hij and metric tensor
gij of F n are defined respectively as:
li =∂F
∂yi, hij = F
∂2F
∂yi∂yjand gij =
1
2
∂2F 2
∂yi∂yj. (3.1)
Let F be a Finsler metric defined by F = m√A, where A is given by A :=
ai1i2...im(x)yi1yi2 ...yim , with ai1...im symmetric in all its indices, Shimada (1979).
Then F is called an m-th root Finsler metric. Clearly, A is homogeneous of degree
m in y.
Let
Ai = aii2...im(x)yi2 ...yim =
1
m
∂A
∂yi, (3.2)
Aij = aiji3...im(x)yi3 ...yim =
1
m(m− 1)
∂2A
∂yi∂yj, (3.3)
Aijk = aijki4...im(x)yi4 ...yim =
1
m(m− 1)(m− 2)
∂3A
∂yi∂yj∂yk. (3.4)
The normalized supporting element of F n is given by
li := Fyi =∂F
∂yi=∂ m√A
∂yi=
1
m
∂A∂yi
Am−1m
=Ai
Fm−1. (3.5)
Consider the transformation
F = F + β, (3.6)
where F = m√A is an m-th root metric and β(x, y) = bi(x)y
i is a one form
on the manifold M . Clearly F is also a Finsler metric on M . Throughout the
chapter we call the Finsler metric F as the Randers transformed m-th root metric
and (M,F ) = Fn as Randers transformed Finsler space. We restrict ourselves
for m > 2 throughout the chapter and also the quantities corresponding to the
Randers transformed Finsler space F n will be denoted by putting bar on the top
of that quantity.
40
3.3 Fundamental metric tensor of Randers trans-
formed m-th root metric
A Finsler metric F = α + β, where α =√aijyiyj is a Riemannian metric and
β = bi(x)yi is a differential one form, was introduced by physicist Randers (1941)
from view point of general theory of relativity. Further Antonelli et al. (1993)
and Matsumoto (1986) studied this metric as a Finsler metric and investigted
its properties. In 1971, M. Matsumoto introduced a Finsler metric F (x, y) =
F (x, y)+β(x, y), where F is a Finsler metric and β is a one form on the manifold
M . This metric is called Randers change of Finsler metric.
The differentiation of (3.6) with respect to yi yields the normalized supporting
element li given by
li = li + bi. (3.7)
In view of (3.5), we have
li =Ai
Fm−1+ bi. (3.8)
Again differentiation of (3.8) with respect to yj yields
hij = (m− 1)F
F
(Aij
Fm−2− AiAj
F 2(m−1)
). (3.9)
From (3.8) and (3.9), the fundamental metric tensor gij of Finsler space F n is
given by:
gij = hij + lilj
gij = (m− 1)F
F
(Aij
Fm−2− AiAj
F 2(m−1)
)+
(Ai
Fm−1+ bi
)(Aj
Fm−1+ bj
)gij =
(m− 1)τAij
Fm−2+Aibj + AjbiFm−1
+ bibj +(1−m− 1τ
) AiAj
F 2(m−1), (3.10)
where τ = FF.
The contravariant metric tensor gij of Finsler space F n is given by, Srivastava and
Arora (2012)
gij =1
τ(m− 1)
[Fm−2Aij − 1
1 + q
biyj + bjyi
F+b2 +m− 1τ − 1
(1 + q)2yiyj
F 2
].
41
Thus contravariant metric tensor gij is rewritten in the form
gij =1
τ(m− 1)
[Fm−2Aij − biyj + bjyi
F+b2 +m− 1τ − 1
F2 yiyj
], (3.11)
where q = βF, (1 + q) = F
F= τ , and matrix (Aij) denotes inverse of (Aij), Yu and
You (2010). Here we have used AijAj = Ai = yi.
Proposition 3.1 The covariant metric tensor gij and contravariant metric
tensor gij of Randers transformed m-th root Finsler space F n are given as :
gij =(m− 1)τAij
Fm−2+Aibj + AjbiFm−1
+ bibj + (1−m− 1τ)AiAj
F 2(m−1)
and
gij =1
τ(m− 1)
[Fm−2Aij − biyj + bjyi
F+b2 +m− 1τ − 1
F2 yiyj
].
3.4 Spray coefficients of Randers transformed m-
th root metric
The geodesics of a Finsler space F n are given by the following system of equations
d2xi
dt2+Gi
(x,dx
dt
)= 0,
where
Gi =1
4gil[F 2]xkyl
yk −[F 2]xl
(3.12)
are called the spray coefficients of F n.
Two Finsler metrics F and F on a manifold M are called projectively related if
there is a scalar function P (x, y) defined on TM0 such that Gi= Gi +Pyi, where
Gi and Gi are the geodesic spray coefficients of F n and F n respectively. In other
words two metrics F and F are called projectively related if any geodesic of the
first is also geodesic for the second and vice versa.
In view of equation (3.10), the metric tensor gij of F n can be rewritten as [Yu and
You (2010)]
gij = τgij +Aibj + AjbiFm−1
+ bibj + (1− τ)AiAj
F 2(m−1), (3.13)
42
where
gij = (m− 1)Aij
Fm−2− (m− 2)
AiAj
F 2(m−1). (3.14)
Further in view of equation (3.11), the contravariant metric tensor gij can be
rewritten as [Yu and You (2010)]
gij =1
τ
[gij − biyj + bjyi
(m− 1)F+
b2 +m− 1τ − 1
(m− 1)F2 − (m− 2)
(m− 1)F 2
yiyj
], (3.15)
where
gij =Fm−2
(m− 1)Aij +
(m− 2)
(m− 1)
yiyj
F 2. (3.16)
The Spray coefficients of Randers transformed Finsler space F n are given by
Gi=
1
4gil[F
2]xkyl
yk −[F
2]xl
.
It can also be written as
Gi=
1
4gil(
2∂gjl∂xk
−∂gjk∂xl
)yjyk
. (3.17)
From (3.13), (3.15) and (3.17), we get
Gi=
1
4τ
[gil + yi
(ϕyl − bl
(m− 1)F
)− biyl
(m− 1)F
]×[(
2∂
∂xk
τgjl +
Ajbl + AlbjFm−1
+ bjbl + (1− τ)AjAl
F 2(m−1)
− ∂
∂xl
τgjk +
Ajbk + AkbjFm−1
+ bjbk + (1− τ)AjAk
F 2(m−1)
)yjyk
],
where
ϕ =b2 + (m− 1)τ − 1
(m− 1)F2 − (m− 2)
(m− 1)F 2.
That is,
Gi=
1
4τ
[gil + yi
(ϕyl − bl
(m− 1)F
)− biyl
(m− 1)F
]×[(
2τ∂gjl∂xk
− τ∂gjk∂xl
+ 2τkgjl − τlgjk
)yjyk + 2
∂
∂xk(Xjl)y
jyk − ∂
∂xl(Xjk)y
jyk], (3.18)
43
where τk = ∂τ∂xk and
Xjl =Ajbl + AlbjFm−1
+ bjbl + (1− τ)AjAl
F 2(m−1). (3.19)
Equation (3.18) can be rewritten as
Gi=
1
4gil(2∂gjl∂xk
− ∂gjk∂xl
)yjyk +
1
4τgil(2τkgjl + 2
∂
∂xk(Xjl)− τlgjk −
∂
∂xl(Xjk)
)yjyk
+1
4τyi[
2τ∂gjl∂xk
+ 2τkgjl + 2∂
∂xk(Xjl)− τ
∂gjk∂xl
− τlgjk −∂
∂xl(Xjk)
yjyk
]×(ϕyl − bl
(m− 1)F
)− biylyjyk
4τ(m− 1)F
2τ∂gjl∂xk
+ 2τkgjl + 2∂
∂xk(Xjl)− τ
∂gjk∂xl
− τlgjk −∂
∂xl(Xjk)
.
Further since
Gi =1
4gil(
2∂gjl∂xk
− ∂gjk∂xl
)yjyk
,
we obtain
Gi= Gi +
1
4τgil(2τkgjl + 2
∂
∂xk(Xjl)− τlgjk −
∂
∂xl(Xjk)
)yjyk
+1
4τyi[
2τ∂gjl∂xk
+ 2τkgjl + 2∂
∂xk(Xjl)− τ
∂gjk∂xl
− τlgjk −∂
∂xl(Xjk)
yjyk
]×(ϕyl − bl
(m− 1)F
)− biylyjyk
4τ(m− 1)F
2τ∂gjl∂xk
+ 2τkgjl + 2∂
∂xk(Xjl)− τ
∂gjk∂xl
− τlgjk −∂
∂xl(Xjk)
.
(3.20)
Now in view of (3.19), we get
Xjkyjyk =
[Ajbk + Akbj
Fm−1+ bjbk + (1− τ)
AjAk
F 2(m−1)
]yjyk.
That is,
Xjkyjyk =
2Aβ
Fm−1+ β2 +
(1− τ)A2
F (2m−2)
= 2Fβ + β2 + (1− τ)F 2
= 2Fβ + β2 + (F − F )F
= 2Fβ + β2 − βF = β(β + F )
= βF
= τβF.
44
Here we have used:
Ajyj = A, bjy
j = β, A = Fm, τ =F
Fand F = F + β.
Substituting the value of gil from equation (3.16) and the value of Xjkyjyk from
above in equation (3.20) we obtain
Gi= Gi +
1
4τ
Fm−2
(m− 1)Ail
[2τ0gjly
j + 2∂
∂xk(Xjl)y
jyk − τlF2 − ∂
∂xl(τβF )
]+yi
4τ
(m− 2)
(m− 1)
yk
F 2
τkF
2 +∂
∂xk(τβF )
+ yi
[ϕyk
4τ
∂
∂xk(F
2)− blyjyk
4τ(m− 1)F
(2∂gjl∂xk
−∂gjk∂xl
)]− biyk
4τ(m− 1)F
∂
∂xk(F
2).
Now from (3.19)
Xjlyj =
Abl + Alβ
Fm−1+ βbl +
(1− τ)AAl
F 2(m−1).
Substituting this value in above equation, we have
Gi= Gi + yi
[ϕyk
4τ
∂
∂xk(F
2)− blyjyk
4τ(m− 1)F
(2∂gjl∂xk
−∂gjk∂xl
)+
1
4τ
(m− 2)
(m− 1)
yk
F 2
τkF
2 +∂
∂xk(τβF )
]+
1
4τ
Fm−2
(m− 1)Ail
[2τ0gjly
j + 2yk∂
∂xk
Abl + Alβ
Fm−1+ βbl +
(1− τ)AAl
F 2(m−1)
− τlF
2 − ∂
∂xl(τβF )
]− biyk
2τ(m− 1)
∂
∂xk(F ).
The above equation can be rewritten as
Gi= Gi + Pyi +Qi, (3.21)
where
P =ϕyk
4τ
∂(F2)
∂xk− blyjyk
4τ(m− 1)F
(2∂gjl∂xk
−∂gjk∂xl
)+
1
4τ
(m− 2)
(m− 1)F 2ykτkF
2 +∂
∂xk(τβF )
and
Qi =1
4τ
Fm−2
(m− 1)Ail
[2τ0yl + 2yk
∂
∂xk
(Abl + Alβ
Fm−1+ blβ +
(1− τ)AAl
F 2(m−1)
)− τlF
2 − ∂
∂xl(τβF )
]− biyk
2τ(m− 1)
∂(F )
∂xk.
45
Further since
Abl + Alβ
Fm−1+ blβ +
(1− τ)AAl
F 2(m−1)=FmblFm−1
+ylF
m−1
F
β
Fm−1+ blβ +
(1− τ)Fm
F 2(m−1)
ylFm−1
F
= Fbl +βylF
+ blβ + (1− τ)yl = (F + β)bl = Fbl,
we have
Qi =1
4τ
Fm−2
(m− 1)Ail
[2τ0yl + 2yk
∂
∂xk(Fbl)− τlF
2 − ∂
∂xl(τβF )
]− biyk
2τ(m− 1)
∂(F )
∂xk.
Now, in view of (3.21), the Finsler metrics F and F are projectively related if
Qi = 0, that is
Fm−2Ail
[2τ0yl + 2yk
∂
∂xk(Fbl)− τlF
2 − ∂
∂xl(τβF )
]= 2biyk
∂(F )
∂xk. (3.22)
Theorem 3.2 The Randers transformed m-th root metric F and m-th root
metric F , on an open subset U ⊂ Rn, are projectively related if equation (3.22) is
satisfied.
3.5 Conformally related Randers transformed m-
th root metric
The conformal transformation between two Finsler metrics F and F are defined
by F (x, y) = eσ(x)F (x, y), where σ is a scalar function on M . We call such two
metrics F and F conformally related. The conformal change is called homothetic
and isometry if σi = ∂σ∂xi = 0 and σ(x) = 0, respectively.
In this section, we prove that if a Randers transformed m-th root metric is con-
formal to a m-th root Finsler metric, then both of them reduce to Riemannian
metrics. More precisely, we prove the following.
Theorem 3.3 Let F = F + β and F = A1m are Randers transformed m-th
root metric and m-th root Finsler metric on an open subset U ⊂ Rn, respectively,
where A := ai1i2...im(x)yi1yi2 ...yim and β = bi(x)y
i. Suppose that F is conformal
to F . Then F and F reduce to Riemannian metrics.
46
In present section, we prove a generalized version of Theorem (3.3). We consider
two Randers transformed m-th root metrics F = F + β and F = F + β, where β
and β are two one forms on M given by β = bi(x)yi and β = bi(x)y
i, which are
conformal. Then we prove the following:
Theorem 3.4 Let F = F + β and F = F + β are two Randers transformed
m-th root Finsler metrics on an open subset U ⊂ Rn, where β = bi(x)yi and
β = bi(x)yi. Suppose that F is non-isometric conformal to F , then F = A
1m is a
Riemannian metric.
Proof: Since F is conformal to F , we have
F = eσF , (3.23)
where F = F + β and F = F + β are Randers transformed m-th root Finsler
metrics on an open subset U ⊂ Rn, with β = bi(x)yi and β = bi(x)y
i.
Then,
F2= e2σF 2.
The metric tensors gij and gij of the two Finsler spaces are related by
gij = e2σgij. (3.24)
Further since F = F + β, in view of equation (3.13) we have
gij = τgij + bibj + (libj + ljbi)−β
Flilj, (3.25)
where τ = FF
and li = Ai
Fm−1 .
Similarly, since F = F + β we have
gij = µgij + bibj +(libj + lj bi
)− β
Flilj, (3.26)
where µ = FF.
Then by (3.24), (3.25) and (3.26), we have
τgij + bibj + (libj + ljbi)−β
Flilj = e2σ
[µgij + bibj +
(libj + lj bi
)− β
Flilj
]
47
gijτ − e2σµ
=(e2σ bibj − bibj
)+(e2σlibj − libj
)(3.27)
+(e2σlj bi − ljbi
)−
(e2σlilj
β
F− lilj
β
F
).
Since σ is not isometry, i.e., σ = 0, then by (3.27), we get
gij =1
τ − e2σµ
(e2σ bibj − bibj
)+(e2σlibj − libj
)(3.28)
+(e2σlj bi − ljbi
)−
(e2σlilj
β
F− lilj
β
F
)which is a function of x alone.
This implies that Cijk = 0 and hence F is Riemannian.
By (3.28), we get the following.
Corollary 3.1 Let F = F + β and F = F + β are two Randers transformed
m-th root Finsler metrics on an open subset U ⊂ Rn, where F = A1m is not
Riemannian, β = bi(x)yi and β = bi(x)y
i. Suppose that F is conformal to F then
F is isometric to F .
Proof of Theorem (3.3) In theorem (3.4), put β = 0 and F = F . Suppose
that the Randers transformed m-th root metric F = F + β is conformal to the
m-th root Finsler metric F = A1m . By Theorem (3.4), F is Riemannian and then
Cijk = 0. Since gij = e2σgij then,
gij + bibj + (libj + ljbi) + (gij − lilj)β
F= e2σgij.
gij
(1− e2σ +
β
F
)= −bibj − (libj + ljbi) + lilj
β
F.
gij =−bibj − (libj + ljbi) + lilj
βF(
1− e2σ + βF
) (3.29)
which yields
gij = e2σ−bibj − (libj + ljbi) + lilj
βF(
1− e2σ + βF
) (3.30)
which is a function of x alone.
Hence
C ijk = Cijk = 0.
Thus C ijk = 0, which implies that F reduces to a Riemannian metric. This
completes the proof.
48
3.6 Locally dually flatness of Randers transformed
m-th root metric
The notion of dually flat Riemannian metrics was introduced by S.-I. Amari and H.
Nagaoka, when they studied the information geometry on Riemannian manifolds.
In Finsler geometry, Z. Shen extended the notion of locally dually flatness for
Finsler metrics. Dually flat Finsler metrics form a special and valuable class of
Finsler metrics in Finsler information geometry, which plays a very important
role in studying flat Finsler information structure. Information geometry has
emerged from investigating the geometrical structure of a family of probability
distributions.
A Randers transformed Finsler metric F = F (x, y) on a manifold M is said to be
locally dually flat, if at any point there is a standard coordinate system (xi, yi) in
TM such that[F
2]xkyl
yk = 2[F
2]xl
. In this case, the coordinate (xi) is called
an adapted local coordinate system. Every locally Minkowskian metric is locally
dually flat.
Consider the Randers transformation F = F +β, where F is an m-th root metric.
We have[F
2]xl=[(F + β)2
]xl = 2 [F + β]
[1
mA
1−mm Axl + βl
]=
2
mA
2−mm Axl + 2A
1mβl +
2
mA
1−mm βAxl + 2ββl. (3.31)
If we put bij = ∂bi∂xj , we have βj = ∂β
∂xj = bijyj.
From (3.31), we get[F
2]xk
=2
mA
2−mm Axk + 2A
1mβk +
2
mA
1−mm βAxk + 2ββk
and [F
2]xkyl
=2
mA
2−mm Axkyl +
2
m
(2−m
m
)A
2−2mm AylAxk (3.32)
+2βk1
mA
1−mm Ayl + 2A
1m blk +
2
mβAxk
(1−m
m
)A
1−2mm Ayl
+2
mβA
1−mm Axkyl +
2
mA
1−mm Axkbl + 2blβk + 2βblk.
49
Let the Finsler metric F is locally dually flat, we have[F
2]xkyl
yk − 2[F
2]xl= 0. (3.33)
Therefore from (3.31), (3.32) and (3.33), we obtained[F
2]xkyl
yk − 2[F
2]xl=
[2
mA
2−mm Axkyl +
2
m
(2−m
m
)A
2−2mm AylAxk
+2βk1
mA
1−mm Ayl + 2A
1m blk +
2
mβAxk
(1−m
m
)A
1−2mm Ayl
+2
mβA
1−mm Axkyl +
2
mA
1−mm Axkbl + 2blβk + 2βblk
]yk
−2
[2
mA
2−mm Axl + 2A
1mβl +
2
mA
1−mm βAxl + 2ββl
]= 0.
The above equation can be written as
Axl
[4
mA
2−mm +
4
mA
1−mm β
]=
2
mA
2−mm A0l +
2
m
(2−m
m
)A
2−2mm A0Ayl
+2βk1
mA
1−mm Ayly
k + 2A1mβl +
2
mβA0
(1−m
m
)A
1−2mm Ayl
+2
mβA
1−mm A0l +
2
mA
1−mm A0bl + 2blβky
k + 2ββl − 4A1mβl − 4ββl,
that is,
Axl
4
mA
1−mm [β + F ] =
2
mA0AylA
1−2mm
(2−m
m
)F + β
(1−m
m
)+
2
mA0lA
1−mm [F + β] + 2βk
1
mA
1−mm Ayly
k
+2
mA
1−mm A0bl + 2blβky
k − 2βl (F + β) .
Therefore F is locally dually flat metric iff
Axl =A0Ayl
2A
[F
mF+
(1−m
m
)]+A0l
2+βkAyly
k
2F+A0bl
2F+mblβky
k
2FA1−mm
− mβl
2A1−mm
.
Thus we have
Theorem 3.5 Let F be a Randers transformed m-th root Finsler metric on
a manifold M . Then, F is locally dually flat metric iff
Axl =A0Ayl
2A
[F
mF+
(1−m
m
)]+A0l
2+βkAyly
k
2F+A0bl
2F+mblβky
k
2FA1−mm
− mβl
2A1−mm
.
50
Chapter 4On Conformal Transformation of m-th root Finsler metric
The purpose of the present chapter is to study the conformal transformation of
m-th root Finsler metric. The spray coefficients, Riemann curvature and Ricci
curvature of conformally transformed m-th root metrics are shown to be certain
rational functions of direction. Further under certain conditions it is shown that
a conformally transformed m-th root metric is locally dually flat if and only if
the transformation is a homothety. Moreover the conditions for the transformed
metrics to be Einstein and Isotropic mean Berwald curvature are also found.
4.1 Introduction
The m-th root Finsler metric has been developed by Shimada (1979), applied to
Biology by Antonelli et al. (1993) and studied by several authors [Srivastava and
Arora (2012), Tayebi et al. (2012), Tayebi and Najafi (2011), Yu and You (2010)
and Prasad and Dwivedi (2002)]. In dimension four, a special fourth root metric
in the form F = 4√y1y2y3y4 is called the Berwald-Moor metric. This metric has
been studied by Balan (2010) and Balan (2006) and considered by physicists as
an important subject for a possible model of space time. Yu and You (2010) have
shown that an m-th root Einstein Finsler metric is Ricci-flat.
The conformal theory of Finsler metric, based on the theory of Finsler spaces
by Matsumoto (1986), has been developed by M. Hashiguchi. Let F and F be
two Finsler metrics on a manifold M such that F = eσ(x)F , where σ is a scalar
function on M , then we call such two metrics F and F are conformally related.
51
More precisely Finsler metric F is called conformally transformed Finsler metric.
A Finsler metric, which is conformally related to a Minkowski metric, is called
conformally flat Finsler metric. The conformal change is said to be a homothety
if σ is a constant.
4.2 Preliminaries
Let F n = (M,F ) be a Finsler space, where M is an n-dimensional C∞-manifold
and F is a Finsler metric.
The normalized supporting element li and angular metric tensor hij of F n are
defined respectively as:
li =∂F
∂yi, hij = F
∂2F
∂yi∂yj. (4.1)
Let F be a Finsler metric defined by F = m√A, where A is given by A :=
ai1i2...im(x)yi1yi2 ...yim , with ai1...im symmetric in all its indices, Shimada (1979).
Then F is called an m-th root Finsler metric. Clearly, A is homogeneous of degree
m in y.
Let
Ai = aii2...im(x)yi2 ...yim =
1
m
∂A
∂yi, (4.2)
Aij = aiji3...im(x)yi3 ...yim =
1
m(m− 1)
∂2A
∂yi∂yj. (4.3)
The normalized supporting element li of F n is given by
li := Fyi =∂F
∂yi=∂ m√A
∂yi=
1
m
∂A∂yi
Am−1m
=Ai
Fm−1. (4.4)
Consider the conformal transformation
F (x, y) = eσ(x)F (x, y)
of m-th root metric F = m√A. Clearly F is also an m-th root Finsler metric on M .
Throughout the chapter we call the Finsler metric F as to conformally transformed
m-th root metric and (M,F ) = Fn as conformally transformed Finser space.
We restrict ourselves for m > 2 throughout the chapter and also the quantities
52
corresponding to the transformed Finsler space F n will be denoted by putting bar
on the top of that quantity, for instance,
A = emσA, Ai = emσAi and Aij = emσAij.
4.3 Fundamental tensor and Spray coefficients of
conformally transformed m-th root metric
The fundamental metric tensor gij of Finsler space F n is given by
gij =1
2
∂2F 2
∂yi∂yj= FFyiyj + FyiFyj .
In view of (4.2), (4.3) and (4.4), we have
gij = (m− 1)Aij
Fm−2− (m− 2)
AiAj
F 2(m−1). (4.5)
The contravariant metric tensor gij of Finsler space F n is given by
gij =Fm−2
(m− 1)Aij +
(m− 2)
(m− 1)
yiyj
F 2, (4.6)
where matrix (Aij) denotes inverse of (Aij), Yu and You (2010). Here we have
used AijAj = Ai = yi.
Since the covariant and contravariant metric tensor of transformed Finsler space
Fn are given by gij = e2σgij and gij = e−2σgij, we have
Theorem 4.1 The covariant metric tensor gij and contravariant metric ten-
sor gij of transformed m-th root Finsler space F n are given as
gij = e2σ((m− 1)
Aij
Fm−2− (m− 2)
AiAj
F 2(m−1)
)(4.7)
and
gij = e−2σ
(Fm−2
(m− 1)Aij +
(m− 2)
(m− 1)
yiyj
F 2
). (4.8)
The geodesics of F n are characterized by a system of equations
d2xi
dt2+Gi
(x,dx
dt
)= 0,
53
where
Gi =1
4gil[F 2]xkyl
yk −[F 2]xl
(4.9)
are called the spray coefficients of F n.
The spray coefficients Gi of F n can be written as
Gi =1
4e−2σgil
∂2(e2σF 2)
∂xk∂ylyk − ∂(e2σF 2)
∂xl
=
1
4e−2σgil
e2σ(F 2
xkylyk − F 2
xl) + 2FFyle2σ2σxkyk − F 2e2σ2σxl
,
i. e.,
Gi = Gi +1
2gil2FFylσxkyk − F 2σxl
, (4.10)
where Gi are given, Yu and You (2010)
Gi =Ail
2(m− 1)
∂Al
∂xkyk − 1
m
∂A
∂xl
. (4.11)
Further in view of equation (4.8) we have
Gi = Gi +1
2
Fm−2
(m− 1)Aij +
(m− 2)
(m− 1)
yiyj
F 2
2Fljσxkyk − F 2σxj
= Gi+
1
2
2F (m−1)yi
(m− 1)Fm−1σxkyk − Fm
(m− 1)Aijσxj +
(m− 2)
(m− 1)
yiyj
F2ljσxkyk − (m− 2)
(m− 1)yiyjσxj
= Gi +
1
2
2yi
(m− 1)σxkyk +
(m− 2)
(m− 1)yiσxjyj − Fm
(m− 1)Aijσxj
.
Here, we have used
Aijlj =yi
Fm−1.
Thus
Gi = Gi +1
2(m− 1)
mσxjyiyj − AAijσxj
. (4.12)
Hence we have
Proposition 4.2 The spray coefficients Gi of the transformed Finsler space
Fn are given by (4.12), where Gi are spray coefficients of Finsler space F n.
In view of equation, (4.11) Gi are rational functions of y, Yu and You (2010).
Hence from equation (4.12), we have
Corollary 4.1 The spray coefficients Gi of the transformed Finsler space F n
are rational functions of y.
54
4.4 Locally dually flat conformally transformed m-
th root metric
Amari and Nagaoka (2000) introduced the notion of dually flat Riemannian met-
rics when they studied the information geometry on Riemannian manifolds. In
Finsler geometry, Shen (2006) extended the notion of locally dually flatness for
Finsler metrics. Dually flat Finsler metrics form a special and valuable class of
Finsler metrics in Finsler information geometry, which play a very important role
in studying flat Finsler information structure. Information geometry has emerged
from investigating the geometrical structure of a family of probability distributions
and has been applied successfully to various areas including statistical inference,
control system theory and information theory [Amari and Nagaoka (2000) and
Amari (1985)].
Definition 4.1 A transformed Finsler metric F = F (x, y) on a manifold M
is said to be locally dually flat, if at any point there is a standard coordinate system
(xi, yi) in TM such that[F
2]xkyl
yk = 2[F
2]xl.
In this case, the coordinate (xi) is called an adapted local coordinate system, Shen
(2006).
For instance, every locally Minkowskian metric is locally dually flat.
Consider the conformal transformation F = eσF , where F is an m-th root metric.
Since
F2
xk = 2e2σσkF2 + e2σF 2
xk = e2σ[F 2xk + 2F 2σk
],
where
σk :=∂σ
∂xk.
We have
F2
xkyl = e2σ[F 2xkyl + 2F 2
ylσk]
F2
xkylyk = e2σ
[F 2xkyly
k + 2Fllσkyk].
55
Therefore
2F2
xl − F2
xkylyk = e2σ
[2F 2
xl + 4F 2σl − F 2xkyly
k − 2ylσ0],
where σ0 := σkyk.
Thus F is locally dually flat metric iff
A(2−2m)
m
m(
2
m− 1)AlA0 + AA0l − 2AAxl
+m
(σ0yl − 2A
2mσl
)= 0, (4.13)
where A0 := Axkyk and A0l := Axkylyk.
The equation (4.13) can be rewritten as
Axl =1
2A
[m(
2
m− 1)AlA0 + AA0l
+m
(σ0yl − 2A
2mσl
)A
(2m−2)m
]. (4.14)
Theorem 4.3 Let F be a conformally transformed m-th root Finsler metric
on a manifold M . Then, F is locally dually flat metric iff (4.14) holds.
Corollary 4.2 If F is locally dually flat metric then the conformally trans-
formed m-th root Finsler metric F is also locally dually flat iff conformal trans-
formation is homothetic.
Proof: In view of Yu and You (2010), F is locally dually flat iff
Axl =1
2A
[(2
m− 1)AlA0 + AA0l
].
Hence F is locally dually flat iff
σ0yl − 2A2mσl = 0. (4.15)
Contracting by yl, we have
σ0F2 − 2F 2σ0 = 0,
i.e. σ0 = 0.
Hence from equation (4.15), σl = 0, i.e. ∂σ∂xl = 0. So σ is constant. Hence the
transformation is homothetic.
The converse is trivial.
56
4.5 Conformally transformed Einstein m-th root
metric
In Finsler geometry, the flag curvature is an analogue of sectional curvature in
Riemannian geometry. A natural problem is to study and characterize Finsler
metrics of constant flag curvature. There are only three local Riemannian metrics
of constant sectional curvature, up to a scaling. However there are lots of non-
Riemannian Finsler metrics of constant flag curvature. For example, the Funk
metric is positively complete and non-reversible with K = −14
and the Hilbert-
Klein metric is complete and reversible with K = −1 [Funk (1929) and Chern and
Shen (2004)].
For a Finsler metric F , the Riemann curvature Ry : TxM → TxM is defined by
Ry(u) = Ri
k(x, y)uk ∂∂xi , u = uk ∂
∂xi , where
Ri
k = 2∂G
i
∂xk− yj
∂2Gi
∂xj∂yk+ 2G
j ∂2Gi
∂yj∂yk− ∂G
i
∂yj∂G
j
∂yk. (4.16)
The Finsler metric F is said to be of scalar flag curvature if there is a scalar
function K = K(x, y) such that
Ri
k = K(x, y)F2δik −
F ykyi
F
. (4.17)
Moreover F is said to be of constant flag curvature if K in equation (4.17) is
constant.
The Ricci curvature of a transformed Finsler metric F on a manifold is a scalar
function Ric : TM → R, defined to be the trace of Ry, i.e.,
Ric(y) := Rk
k(x, y)
satisfying the homogeneity Ric(λy) = λ2Ric(y), for λ > 0. A Finsler metric F
on an n-dimensional manifold M is called an Einstein metric if there is a scalar
function K = K(x) on M such that
Ric = K(n− 1)F2.
57
A Finsler metric is said to be Ricci-flat if Ric = 0. By formula (4.16) and corollary
(4.1), we get the following
Lemma. Ri
k and Ric = Rkk are rational functions in y.
Proposition 4.4 Let F be a non-Riemannian conformally transformed m-th
root Finsler metric with m > 2 on a manifold M of dimension n > 1. If F is an
Einstein metric, then it is Ricci-flat.
Proof: If F is an Einstein metric, i.e. Ric = K(n− 1)F2, and F 2 is an irrational
function, as m > 2 and Ric are rational function of y. Therefore K = 0 and hence
Ric = 0.
Corollary 4.3 Let F = eσ(x)F be a non-Riemannian transformed m-th root
Finsler metric with m > 2 on a manifold M of dimension n > 1. If F is of
constant flag curvature K, then K = 0.
4.6 Conformally transformed m-th root metric with
Isotropic E-curvature
Let if Gi be spray coefficients of a Finsler space F n then the Berwald curvature of
Fn is defined as
Bi
jkl =∂3G
i
∂yj∂yk∂yl.
A transformed Finsler metric F is called a Berwald metric if spray coefficients Gi
are quadratic in y ∈ TxM , for any x ∈ M or equivalently, the Berwald curvature
vanishes. The E-curvature is defined by the trace of the Berwald curvature, i.e.,
Eij =12B
m
mij. A Finsler metric F on an n-dimensional manifold M is said to be
isotropic mean Berwald curvature or of isotropic E-curvature if
Eij =c(n+ 1)
2Fhij, (4.18)
where hij = gij − gipypgjqy
q is the angular metric and c = c(x) is a scalar function
on M . If c = 0, then F is called weakly Berwald metric.
58
From equation (4.7), we have
gij = e2σgij = e2σ((m− 1)
Aij
Fm−2− (m− 2)
AiAj
F 2(m−1)
). (4.19)
The angular metric is given by
hij = gij − lilj = e2σ((m− 1)
Aij
Fm−2− (m− 1)
AiAj
F 2(m−1)
). (4.20)
From equation (4.18) and (4.20), we have
Eij =(n+ 1)c
2Fe2σ(m− 1)
(Aij
Fm−2− AiAj
F 2(m−1)
)
=(n+ 1)c
2eσ(m− 1)
(Aij
Fm−1− AiAj
F (2m−1)
) (F = eσF
)
=(n+ 1)c
2eσ(m− 1)F
(Aij
A− AiAj
A2
) (F = A
1m
)=
(n+ 1)
2A2A
1m eσ(m− 1)c (AijA− AiAj) . (4.21)
In view of equation (4.12), we see that Eij are rational functions with respect to
y. Thus from equation (4.21), we have either c = 0 or
(AijA− AiAj) = 0. (4.22)
Suppose that c = 0. Contracting (4.22) with Ajk yields
Aδki − Aiyk = 0,
which implies that nA = A. This contradicts our assumption n > 1. Therefore
c = 0 and consequently Eij = 0. Thus we have
Proposition 4.5 If F is of isotropic mean Berwald curvature. Then F is
weakly Berwald metric.
59
Chapter 5Transformation of a Finsler Space byNormalised Semi-Parallel Vector Fields
The purpose of the present chapter is to study the properties of a modified Finsler
space obtained by transformation of a Finsler space with the help of two normalised
semi-parallel vector fields.
5.1 Introduction
The semi-parallel vector field in Riemannian Geometry has been introduced by
Fulton (1965), where as in Finsler geometry by Singh and Prasad (1983), for
instance, torse forming vector fields studied by Yano (1944), concurrent vector
fields studied by Yano (1943) and concircular vector fields studied by Adati (1951)
are semi-parallel. The notations and terminology are referred to monograph of
Matsumoto (1986).
Let F n = (M,F ) be a Finsler space, where M is an n-dimensional C∞-
manifold and F is a Finsler metric. The Cartan tensor Cijk of F n is given by
Cijk = 12
∂gij∂yk
. Let CΓ =(F ijk, N
ik, C
ijk
)be Cartan connection on the Finsler Space
F n such that the horizontal and vertical derivative of a vector field X i is written
as
X i|j =
δXi
δxj +XhF ihj and X i|j = ∂Xi
∂yj+XhC i
hj, where δδxj = ∂
∂xj −Nmj
∂∂ym
.
Definition 5.1 [Kitayama (1998)] A normalised vector field X i(x) in a Finsler
space F n is said to be parallel if:
(i) X i is function of coordinate only,
60
(ii) X i|j := δjX
i +XhF ihj = 0, and
(iii) X i|j := ∂jXi +XhCi
hj = XhCihj = 0,
where ∂j stands for ∂∂xj , ∂j stands for ∂
∂yjand δj stands for δ
δxj .
Definition 5.2 [Pandey and Diwedi (1999) and Singh and Prasad (1983)] A
normalised vector field Xi in a Finsler space F n is said to be semi-parallel if:
(i) Xi is function of coordinate only,
(ii) CijkXi = 0, and
(iii) Xi|j = ρ(gij−XiXj), where ρ is a non-zero scalar function of coordinate only.
Consider two normalised semi-parallel vector fields X(α)i , α = 1, 2 in a Finsler
space F n with scalars ρ(α), α = 1, 2 respectively, that is
(i) X(α)i are functions of coordinate only,
(ii) CijkX
(α)i = 0, and
(iii) X(α)i|j = ρ(α)(gij −X
(α)i X
(α)j ), for α = 1, 2.
Kitayama studied a transformed Finsler space F ∗n with metric F ∗(x, y), given by
F ∗2 = F 2 + (β)2, where β = Xiyi and Xi is a parallel vector field; where as Singh
and Prasad (1983) studied a transformed Finsler space F ∗n with metric F ∗(x, y)
given by, F ∗2 = F 2 + (β)2, where β = Xiyi and Xi is a semi-parallel vector field.
The purpose of the present chapter is to study the properties of a transformed
Finsler space F ∗n with the metric F ∗(x, y), given by
F ∗2 = F 2 + β(1)β(2) (5.1)
where β(1) = X(1)i yi and β(2) = X
(2)i yi are two one forms and X
(α)i ; α = 1, 2; are
semi-parallel vector fields satisfying the condition X(1)i X
(2)j = X
(1)j X
(2)i .
Lemma. [Singh and Prasad (1983)] If a Finsler space F n admits normalised
semi-parallel vecror fields X(α)i ; for α = 1, 2, then there exist no functions ξ(α)(x, y)
such that X(α)i = ξ(α)(x, y)yi.
61
5.2 Transformed Finsler space obtained by Nor-
malised semi-parallel vector fields
Consider the transformed Finsler space F ∗n, whose metric F ∗(x, y) is given by
equation (5.1). Differentiating equation (5.1) with respect to yi, the line element
l∗i of F ∗n is given by
2F ∗l∗i = 2Fli + β(1)X(2)i + β(2)X
(1)i . (5.2)
Again differentiating (5.2) with respect to yj, we have the metric tensor g∗ij of F ∗n
as
g∗ij = gij +X(1)i X
(2)j , (5.3)
where gij is the metric tensor of F n.
The reciprocal metric tensor g∗ij of F ∗n is given by
g∗ij = gij − 1
1 + λX(1)iX(2)j, (5.4)
where λ = X(1)i X(2)i.
Again differentiating (5.3) and using (5.4), we have
C∗ijk = Cijk, C
∗ijk = C i
jk. (5.5)
In view of above equation, if F n is Riemannian then F ∗n is also Riemannian. More-
over the (h)hv torsion tensor is invariant under the transformation (5.1). From
definition (5.2) and from Ricci identities for h and v− covariant differentiations
we have
RhijkX(α)h =
ρk + (ρ(α))2X
(α)k
gij−
ρj + (ρ(α))2X
(α)j
gik−X(α)
i
ρkX
(α)j − ρjX
(α)k
,
PhijkX(α)h = ρ(α)Cijk, (5.6)
and
ShijkX(α)h = 0,
where Rhijk, Phijk and Shijk are the components of the h-curvature tensor, hv-
curvature tensor and v-curvature tensor respectively and ρ(α)i = ∂ρ(α)
∂xi . From (5.3),
62
(5.5) and definition (5.2), we have
F ∗ijk = F i
jk +1
2(1 + λ)
[ρ(2)X(1)i
(gjk −X
(2)j X
(2)k
)+ ρ(1)X(2)i
(gjk −X
(1)j X
(1)k
)],
(5.7)
where F ∗ijk and F i
jk are the Cartan’s connection parameters of F ∗n and F n respec-
tively.
From definition (5.2), equation (5.3), (5.4), (5.5) and (5.7), we get the three Car-
tan’s curvature tensors, given by
S∗hijk = Shijk and P ∗
hijk = Phijk and
R∗hijk =Rhijk +
ρ(1)ρ(2)
2(1 + λ)[2(ghjgik − ghkgij) (5.8)
+ ghk
(λ+ 1)X
(1)i X
(1)j +X
(2)i X
(2)j −X
(1)i X
(2)j
− ghj
X
(1)i X
(1)k (λ+ 1) +X
(2)i X
(2)k −X
(1)i X
(2)k
+ gij
(X
(1)h X
(1)k +X
(2)h X
(2)k
)− gik
(X
(1)h X
(1)j +X
(2)h X
(2)j
)]+
1
4(1 + λ)2
[a(λ+ 1)X(1)i + b(X(1)i +X(2)i)
(ghkX
(1)j − ghjX
(1)k )
+c(λ+ 1)X(1)i + d(X(1)i +X(2)i)
(ghkX
(2)j − ghjX
(2)k )]+
λ
λ+ 1
X
(1)i X(2)
m Rmhjk
,
where a = (ρ(2))2 − 2λρ(1)ρ(2), b = 2ρ(1)ρ(2) − λ(ρ(1))2,
c = 2ρ(1)ρ(2) − λ(ρ(2))2 and d = (ρ(1))2 − 2λρ(1)ρ(2).
Theorem 5.1 Let a Finsler space F n admits normalised semi-parallel vecror
fields X(α)i and F ∗n be a modified Finsler space given by (5.1). Then hv and
v-curvature tensors of F n and F ∗n are identical but the h-curvature tensors are
related by (5.8).
Contracting (5.7) by yjyk, we have
G∗i = Gi +1
4(1 + λ)
[ρ(2)X(1)i
F 2 − (β(2))2
+ ρ(1)X(2)i
F 2 − (β(1))2
]. (5.9)
Differentiating (5.9) with respect to yj and yk respectively, we have
G∗ijk = Gi
jk +1
2(1 + λ)
ρ(2)X(1)i
(gjk −X
(2)j X
(2)k
)+ ρ(1)X(2)i
(gjk −X
(1)j X
(1)k
),
(5.10)
63
where
2Gi = F ijky
jyk, X0 = Xjyj and Gi
jk =∂2Gi
∂yj∂yk
are the Berwald’s connection parameters.
Theorem 5.2 If the vector field X(1)i is parallel and X(2)
i is semi-parallel in
Finsler space F n then the vector field X(2)i is semi-parallel also in transformed
Finsler space F ∗n obtained under the transformation (5.1).
Proof. In view of definitions (5.1) and (5.2) X(1)i|j = 0, that is, ρ(1) = 0 and
X(2)i|j = ρ(2)(gij−X(2)
i X(2)j ). Further since Cartan tensor Cijk and Ci
jk are invariant
under transformation (5.1). We need only to show thatX(2)i|∗j = k(gij−X(2)
i X(2)j ) for
some scalar k. Here |∗ denotes h-covariant differentiation with respect to Cartan
connection CΓ∗ of F ∗n. Considering (5.7) we have
X(2)i|∗j = X
(2)i|j − X
(2)h
2(1 + λ)
[ρ(2)X(1)h
(gij −X
(2)i X
(2)j
)+ ρ(1)X(2)h
(gij −X
(1)i X
(1)j
)]= k(gij −X
(2)i X
(2)j ),
(5.11)
where k = (2+λ)ρ(2)
2(1+λ).
Corollary. If the vector fields X(1)i and X
(2)i are parallel in Finsler space F n
then these vector fields are also parallel in transformed Finsler space F ∗n obtained
under the transformation (5.1).
Suppose the spaces F n and F ∗n are in geodesic correspondence [Singh and Prasad
(1983)] then
G∗i = Gi + P (x, y)yi, (5.12)
where P (x, y) is positively homogeneous of degree one in yi.
Comparing (5.9) and (5.12), we get
1
4(1 + λ)
[ρ(2)X(1)i
F 2 − (β(2))2
+ ρ(1)X(2)i
F 2 − (β(1))2
]= Pyi. (5.13)
Contracting above by X(1)i and X(2)i separately, we have
P =ρ(2)
(F 2 − (β(2))2
)(λ− 1)
4 (λβ(2) − β(1))=ρ(1)
(F 2 − (β(1))2
)(λ− 1)
4 (λβ(1) − β(2)), (5.14)
provided λ2 = 1. Thus we have
64
Theorem 5.3 If a Finsler space F n admits normalised semi-parallel vector
fields X(α)i and F ∗n be a modified Finsler space given by (5.1). Then the spaces
F n and F ∗n are in geodesic correspondence if P satisfies (5.14) with λ2 = 1.
Remark. If the two normalised semi-parallel vector fields are same, we have
λ = 1. In view of Theorem (2) of Singh and Prasad (1983), the two spaces are not
in geodesic correspondence.
Next suppose that F n and F ∗n are in conformal correspondence that is g∗ij =
ψ(x)gij. Comparing this with equation (5.3), we have
Theorem 5.4 Let a Finsler space F n (n > 2) admits normalised semi-
parallel vecror fields X(α)i and F ∗n be a modified Finsler space given by (5.1).
Then the spaces F n and F ∗n are not in conformal correspondence.
5.3 Special Finsler spaces with semi-parallel vec-
tor fields
In this section we consider some special Finsler spaces admitting normalized semi-
parallel vector fields. First we consider a Finsler space with T-condition, that
is
Thijk := FChij|k + lhCijk + liChjk + ljChik + lkChij = 0. (5.15)
Contracting (5.15) by Xh, we have XhlhCijk = 0 by Definition (5.2) (ii). Thus we
have Cijk = 0, because Xhlh = β(1)
F= 0. Consequently from (5.5), we have
Theorem 5.5 If a Finsler space F n satisfying T-condition admits a semi-
parallel vector field, then both the Finsler spaces are Riemannian.
The generalised T -condition is defined by
Tij := Tijrsgrs = FCi|j + liCj + ljCi = 0, (5.16)
where the tensor Tij is called the contracted T -tensor and Ci = Cijkgjk is the
torsion vector [Tiwari (2012) and Pandey and Tiwari (1999)].
Contracting (5.16) byX i and using Definition (5.2) (ii), we have Cj = 0. According
to Deiche’s theorem and from (5.5), we have
65
Theorem 5.6 If a Finsler space F n satisfying generalised T -condition ad-
mits a semi-parallel vector field, then both the Finsler spaces F n and F ∗n are
Riemannian.
A Finsler space F n, (n > 2) is called quasi C-reducible if the torsion tensor Cijk
is written as
Cijk = AijCk + AjkCi + AkiCj, (5.17)
where Aij is symmetric tensor satisfying Ai0 = Aijyj = 0.
Contracting (5.17) by X iXj we get ξCk = 0, where ξ = AijXiXj. Thus we have
Theorem 5.7 If a quasi C- reducible Finsler space F n, (n > 2) admits semi-
parallel vector fields then the spaces F n and F ∗n are Riemannian provided ξ = 0.
Now we consider a C-reducible Finsler space which is characterized by
Cijk = AijCk + AjkCi + AkiCj, (5.18)
where Aij =hij
(n+1)from Matsumoto (1972). Since hijX iXj = 0 for a Finsler space
with n > 2.
Corollary. If a C- reducible Finsler space F n, (n > 2) admits semi-parallel vector
fields then the spaces F n and F ∗n are Riemannian.
An n-dimensional Finsler space F n is said to be semi-C-reducible if its (h)hv-
torsion tensor Cijk is written as
Cijk = p[hijCk + hjkCi + hkiCj] + q[CiCjCk], (5.19)
where p, q are scalars satisfying p(n+1)+qC2 = 1 and C2 = gijCiCj. In particular
if p = 0 but C2 = 0, that is, the torsion tensor Cijk is written as Cijk =CiCjCk
C2 ,
F n is said to be C2 − like. Contracting (5.19) by X iXj we have pCk = 0 which
shows p = 0, thus we have
Theorem 5.8 If a semi-C- reducible Finsler space F n, (n > 2) admits semi-
parallel vector fields then the spaces F n and F ∗n are C-2 like.
66
A P-2 like Finsler space F n (n > 2) is characterized by
Phijk = KhCijk −KiChjk, (5.20)
where Kh is a covariant vector field. Contracting above equation by X(α)h and
using (5.6) and Definition (5.2) (ii), we have (X(α)hKh − ρ(α))Cijk = 0. Thus we
have
Theorem 5.9 If a P2-like Finsler space F n, (n > 2) admits semi-parallel
vector fields then the spaces F n and F ∗n are Riemannian provided X(α)hKh = ρ(α)
for any α = 1, 2.
An n-dimensional Finsler space F n is called a Landsberg space if the (v)hv-torsion
tensor Pijk of F n vanishes. Further a Finsler space F n is called P-reducible if
torsion tensor Pijk of F n is written in the form
Pijk =1
n+ 1(hijPk + hjkPi + hkiPj), (5.21)
where Pi = P rir = Ci|0. Contracting above equation by X iXj and using (5.6), we
obtain hijX iXjPk = 0. But since hijX iXj = 0 we have Pk = 0. Thus
Theorem 5.10 If a P-reducible Finsler space F n, (n > 2) admits semi-
parallel vector fields then the spaces F n and F ∗n are Landsberg.
67
Chapter 6Predator-prey model with prey refuges:Jacobi stability vs Linear stability
In this chapter, the dynamics of a predator-prey model is proposed and analyzed.
Three types of refuges: those that protect a constant number of prey population, a
constant proportion of prey population and a function of predator-prey encounters
using refuges are considered. Linear stability analysis based on Lyapunov theory
and Jacobi stability analysis based on KCC theory are carried out. Comparisons
of results obtained in both cases shows that, Jacobi stability analysis of these
models reflects the better ecological interpretation.
6.1 Introduction
Geometry is the link between physical world and its visualization. Nature’s ge-
ometrisation has been the subject of theoretical interest. This contains use of
geometric hypothesis and applications in the natural sciences, such as the general
theory of relativity and Cosmology in physics [Yamasaki and Yajima (2013)]. In
biology, nature’s geometrisation is also of scientific and technological interest. For
example, the KCC theory which are applied to many biological problems, such
as production in the Volterra method (Antonelli et al. (2003) and Antonelli et
al. (1993)), the Volterra-Hamilton system ( Antonelli et al. (2011) and Antonelli
and Bucataru (2001)), Tyson’s model for the cell division cycle (Antonelli et al.
(2002)) and the robustness of biological systems (Sabau (2005)). The notion of the
KCC (Kosambi-Cartan-Chern) theory regard in works of Kosambi (1933), Cartan
(1933) and Chern (1939), and the abbreviation KCC come from the names of
68
these three initiators, who profound the geometric theory of a system of second
order ordinary differential equation (SODE). The first attempt to establish and to
develop systematically the KCC theory is due to Antonelli and Bucataru (2001).
The KCC theory describes the evolution of a dynamical system in geometric terms,
by considering it as a geodesic in a Finsler space. Thus a second order differential
equation can be investigated in geometric terms by KCC-theory inspired by the
geometry of a Finsler space. By associating a non-linear connection and a Berwald
type connection to the differential system, five geometrical invariants are obtained
with the second invariant giving the Jacobi stability of the system.
There are so many Mathematicians and Ecologists among them the prey-
predator model is a subject of great interest. The linear and non-linear asymp-
totic stability conditions of model in a homogeneous habitat have been obtained.
It is observed that many species have already become extinct and many others
are at the stage of extinction due to several reasons such as over exploitation,
mismanagement of resources, indiscriminate harvesting, over predation, loss of
habitat and environmental pollution, etc. To save the species from getting ex-
tinction we are taking measures like improving conditions of their habitat, reduce
the interaction of the species with external agents which tend to decrease their
numbers, impose restriction on harvesting, creating refuges, establish protected
areas etc. so that the species grow in these protected areas without any exter-
nal disturbances and hence the protected population can improve their numbers.
Chattopadhyay et al. (2000) studied a prey-predator model with some cover on
prey species. Kar (2006) proposed a predator-prey model incorporating a prey
refuge and independent harvesting on either species. Recently, (Devi (2013) and
Devi (2012)) studied the effect of prey refuge on a ratio-dependent predator-prey
model with stage-structure and non constant harvesting of prey population, re-
spectively. In these models she observed that prey refuge parameter plays a very
crucial role in the analysis.
In the general theory of relativity and Cosmology we can geometrise the inter-
action between masses but in this chapter we geometrise the interaction between
living things. The purpose of present chapter is to consider the differential geo-
69
metric structure for some dynamical systems of the predator-prey model based on
KCC theory.
Organization of this chapter is as follows:
KCC theory and Jacobi Stability theory are described in section 6.2. Section 6.3
presents the applications of geometric theory to second order systems. Mathe-
matical models and their analyses are given in subsection 6.4.1, 6.4.2 and 6.4.3
of section 6.4. Numerical simulations and discussions are presented in section 6.5
followed by conclusion in section 6.6.
6.2 Preliminary
6.2.1 About Kosambi-Cartan-Chern (KCC) theory and Ja-
cobi stability
In present section, we recall the basics of KCC theory and Jacobi stability based
on the papers of Bohmer et al. (2010), Yajima and Nagahama (2007) and Antonelli
and Bucataru (2001).
Let M be a real smooth n-dimensional manifold, and (TM, π,M) be its tangent
bundle, where π : TM −→ M is a projection from the total space TM to the
base manifold M . A point x ∈ M has local coordinates (xi), where i = 1, ..., n.
The local chart of a point in TM is denoted by (xi, xi), where xi = dxdt
and t is an
absolute invariant.
The equations of motion of a dynamical system can be derived from a Lagrangian
L via the Euler-Lagrange equations. For a regular Lagrangian the Euler-Lagrange
equations are equivalent to a system of second-order differential equations
xi +Gi(x, x) = 0, (6.1)
where Gi(x, x) is a smooth function. Vary the trajectories xi(t) of the system (6.1)
into nearby ones according to
xi(t) = xi(t) + ϵui(t), (6.2)
70
where ui(t) are components of a contravariant vector field along the path xi(t)
and ϵ denotes a scalar parameter with small value |ϵ|. Substituting from (6.2) into
equations (6.1) and taking the limit ϵ→ 0 we obtain the variational equations
ui +∂Gi
∂xjuj +
∂Gi
∂xjuj = 0. (6.3)
By using the KCC-covariant differential we can write equation (6.3) in the covari-
ant formD2ui
Dt2= P i
juj, (6.4)
where D(...)/Dt is a covariant differential defined by
Dui
Dt=dui
dt+N i
juj, (6.5)
N ij are coefficient of the non-linear connection given by
N ij =
1
2
∂gi
∂xj, (6.6)
P ij is the deviation curvature tensor
P ij = − ∂gi
∂xj+∂N i
j
∂xkxk −Gi
jkgk +N i
kNkj (6.7)
and Gijk is a Finsler (Berwald) connection given by Antonelli (2003),
Gijk =
∂N ij
∂xk. (6.8)
Equation (6.4) is called the Jacobi equations, or the variation equations attached
to the system of SODE, and P ij is called the second KCC-invariant or the deviation
curvature tensor. When the system (6.1) describes the geodesic equations in either
Riemann or Finsler geometry, equation (6.4) is the usual Jacobi equation.
The first term of (6.7): ∂gi
∂xj is the curvature when Finsler (Berwald) connection
and coefficient of the non-linear connection become zero, we get
Zij =
∂gi
∂xj. (6.9)
The deviation curvature tensor P ij gives the stability of whole trajectories via the
following theorem, Antonelli et al. (1993) :
71
Theorem 6.1 The trajectories of (6.1) are Jacobi stable if and only if the real
parts of the eigenvalues of the deviation curvature tensor P ij are strictly negative
everywhere, and Jacobi unstable, otherwise.
In particular, the trajectories of the one-dimensional system are Jacobi stable
when P 11 ≤ 0, and Jacobi unstable when P 1
1 > 0.
The third, fourth and fifth invariants of the system (6.1) are given by Antonelli
(2003)
P ijk ≡
1
3
(∂P i
j
∂xk− ∂P i
k
∂xj
), P i
jkl ≡∂P i
jk
∂xl, Di
jkl ≡∂Gi
jk
∂xl. (6.10)
The third invariant is a torsion tensor, while the fourth and fifth invariants are the
Riemann-Christoffel curvature tensor, and the Douglas tensor, respectively given
by Antonelli (2003). The second invariant is expressed using the third invariant
and the h−covariant derivative of the first invariant where as the vertical compo-
nent of a semispray with respect to non-linear connection, gives the first invariant
of semispray, called the deviation tensor. The third, fourth and fifth tensors al-
ways exist and they describe the geometrical properties of a system of SODE (6.1)
(Sabau (2005) and Antonelli (2003)).
Throughout this chapter, Einstein’s summation convention is used and we call Zij
as the zero-connection curvature.
6.3 Applications of geometric theory to second or-
der system
In present section, we derive deviation curvature for some dynamical systems of
the predator-prey model. Linear stability analysis are applied for the predator-
prey model, which is the theory of local stability around a point on the tangent
space. In this case, the equation (6.1) is a first-order differential equation with
respect to x and equation (6.3) is the Jacobi stability which reduces to an equation
in a linear stability theory given by Yajima and Nagahama (2008). Therefore, the
Jacobi stability gives a global stability than the linear stability. The Jacobian
matrix of the linearised system plays an important role in linear stability analysis.
72
It would be interesting to correlate linear stability with Jacobi stability. In other
words, to compare the signs of the eigenvalues of the Jacobian matrix J at a
fixed point with the signs of the eigenvalues of the deviation curvature tensor P ij
evaluated at the same point. We express the geometric quantities of KCC theory
in terms of the Jacobian matrix of the linearised system.
6.3.1 Geometric quantities and Jacobian matrix
Let us consider a two-dimensional vector field described by
xi = f i(x), (6.11)
where i = 1, 2, ..., n and f i denote a given function. Equation (6.11) is approx-
imated by a linear system around an equilibrium point xi0 using the relation
xi = xi0 + ηi, where ηi is a small quantity. That is,
ηi = J ij(x0)η
j, (6.12)
where J ij(x0) is the Jacobian matrix of f i evaluated at point x0.
Even though this should be possible in the general case but here we consider the
two-dimensional case (i = 1, 2)
η = J(x0)η, (6.13)
where
η =(η1, η2
)T, (6.14)
J(x0) = J =
∂1f1(x0) ∂2f
1(x0)
∂1f2(x0) ∂2f
2(x0)
(6.15)
and ∂i = ∂(...)/∂xi. The simultaneous differential equation (6.13) can be rewritten
as a SODE. When we consider the coordinate system (ηi, ηi), we have the following
equation for i = 1 :
η1 − tr[J ]η1 + det[J ]η1 = 0. (6.16)
This is a particular case of (6.1) for g1 = −tr[J ]η1+det[J ]η1. Therefore, equations
(6.6), (6.9) and (6.7) give the non-linear connection, the zero-connection curvature
73
and the deviation curvature of the linearised system, respectively as:
N = −1
2tr[J ], (6.17)
Z = det[J ], (6.18)
P =1
4
(tr[J ])2 − 4det[J ]
= N2 − Z, (6.19)
where N = N11 , Z1
1 = Z and P = P 11 (we use same notations throughout the
chapter). From equation (6.8), the Finsler connection vanishes in this linearised
system. Now, equations (6.17), (6.18) and (6.19) show that the geometric quanti-
ties of the linearised system can be easily calculated when the Jacobian matrix of
the system is obtained. The system is
linear stable for N > 0,
linear unstable for N < 0.
The left term of equation (6.19) is related to the Jacobi stability, i.e.
Jacobi stable for P ≤ 0,
Jacobi unstable for P > 0.
6.4 Mathematical models
6.4.1 Predator-prey model for a constant number of prey
using refuges with exponential growth rate
Consider a habitat where prey and predator species are living together. The
predator-prey interaction between any two species occurs when one species (the
predator) feeds on the other species ( the prey ).
In 1925 the American biophysicist Alfred James Lotka and Italian Mathematician
Vito Volterra describe a simple Mathematical model for the interaction between
predators and their prey by means of the following non-linear differential equa-
tions,
x(t) = rx− a(x−m)y, x(0) = x0 > m > 0, (6.20)
74
y(t) = b(x−m)y − cy, y(0) = y0 > 0. (6.21)
Here, x(t) and y(t) are densities of prey and predator population respectively
at time t. r is the natural growth rate of the prey population, m is constant
prey population, a is the rate at which predators capture prey, b is the rate at
which predators increase by consuming prey and c is the natural death rate of the
predator population. All parameters r,m, a, b and c are assumed to be positive.
This system can be approximated by a linear system around an equilibrium point
(x01, y01). From (6.15), the Jacobian matrix of the system is
J =
r − ay01 −a(x01 −m)
by01 b(x01 −m)− c
. (6.22)
Next, we consider the deviation curvature in two cases: (i) when both species
coexist, (ii) when both species extinct.
In the case when the two species coexist, i.e. x01 = 0 = y01, we have equilibrium
point (x01, y01), where x01 = m+ cband y01 = r
ac(bm+c). Therefore, (6.22) becomes
J =
− rbmc
−acb
rbac(bm+ c) 0
. (6.23)
From (6.17) and (6.18), the non-linear connection and the zero-connection curva-
ture are given by N = 12bmrc> 0 and Z = r(bm + c) > 0. Then, from (6.19), the
deviation curvature of coexistence is
P =1
4
b2m2r2
c2− r(bm+ c). (6.24)
Here system is always linear stable because N > 0.
In the case of extinction, i.e. x01 = 0 = y01, (6.22) becomes
J =
r am
0 −bm− c
. (6.25)
In this case, since N = (bm+c−r)2
and Z = −(bm+ c)r < 0, the deviation curvature
of extinction is
P = N2 − Z =
[(bm+ c+ r)
2
]2> 0. (6.26)
This is always positive, i.e. the extinction state is always Jacobi unstable where
as system is linear stable if N > 0, i. e., if bm + c > r, and linear unstable if
bm+ c < r.
75
6.4.2 Predator-prey model with constant proportion of prey
using refuges
Consider the predator-prey model model with constant proportion of prey using
refuges. The standard form of the system is given by
x(t) = rx− a(1− η)xy, x(0) = x0 > 0, (6.27)
y(t) = b(1− η)xy − cy, y(0) = y0 > 0. (6.28)
0 < η < 1,
where η is constant proportion of prey which is assumed to be positive and other
parameters r, a, b and c have same meaning as given in section (6.4.1).
This system can be approximated by a linear system around an equilibrium point
(x02, y02). From (6.15), the Jacobian matrix of the system is
J =
r − ay02 + aηy02 −ax02 + aηx02
by02 − bηy02 bx02 − bηx02 − c
. (6.29)
Next, we consider the deviation curvature in two cases: (i) when both species
coexist, (ii) when both species extinct.
In the case when the two species coexist, i.e. x02 = 0 = y02, we have equilibrium
point (x02, y02), where x02 = cb(1−η)
and y02 = ra(1−η)
. Therefore, (6.29) becomes
J =
0 −acb
bra
0
. (6.30)
From (6.17) and (6.18), the non-linear connection and the zero-connection curva-
ture are given by
N = 0 and Z = cr, respectively. Then, from (6.19), the deviation curvature for
the two species coexisting is
P = −cr. (6.31)
In the case of extinction, i.e. x02 = 0 = y02, (6.29) becomes
J =
r 0
0 −c
. (6.32)
76
In this case, since N = −12(r − c) and Z = −rc, the deviation curvature for the
case of extinction is
P =1
4(r + c)2. (6.33)
For the Jacobi stable, P should be negative or zero, otherwise Jacobi unstable but
here P is always positive. Therefore the extinction state is the Jacobi unstable
where as system is linear stable if N > 0, i. e., if r < c, and linear unstable if
r > c.
6.4.3 Predator-prey model with the function of predator-
prey encounters for prey refuge
Consider the predator-prey model with a function of predator-prey encounters for
prey refuge. The standard form of the system is given by
x(t) = rx− a(1− ϵy)xy, x(0) = x0 > 0, (6.34)
y(t) = b(1− ϵy)xy − cy, y(0) = y0 > 0. (6.35)
where ϵ > 0 is a proportionality constant or predator-prey encounters function and
all other parameters r, a, b and c have same meaning as given in section (6.4.1).
This system can be approximated by a linear system around an equilibrium point
(x03, y03). From (6.15), the Jacobian matrix of the system is
J =
r − a(1− ϵy03)y03 −ax03 + 2ϵax03y03
b(1− ϵy03)y03 bx03 − 2bϵx03y03 − c
. (6.36)
Next, we consider the deviation curvature in two cases: (i) when both species
coexist, (ii) when both species extinct.
In the case when the two species coexist, i.e. x03 = 0 = y03, we have equilibrium
point (x03, y03), where x03 = 2ac
b[a∓√a2−4aϵr]
and y03 = a∓√a2−4aϵr2aϵ
.
Taking positive sign, we get an equilibrium point
(x03, y03) =
(2ac
b[a+
√a2 − 4aϵr
] , a+√a2 − 4aϵr
2aϵ
). (6.37)
77
Therefore, Jacobian matrix (6.36) for equilibrium point (x03, y03) becomes
J =
0 2ac√a2−4aϵr
b(a+√
a2−4aϵr)
bra
−ac−3c√a2−4aϵr
a+√a2−4aϵr
. (6.38)
From (6.17) and (6.18), the non-linear connection and the zero-connection curva-
ture for equilibrium point (x03, y03) are given by
N = 12ac+3c
√a2−4aϵr
a+√a2−4aϵr
and Z = −2cr√a2−4aϵr
a+√a2−4aϵr
, respectively.
Then, from (6.19), the deviation curvature for the two species coexisting is
P =c
4(a+√a2 − 4aϵr)2
[c(a+ 3
√a2 − 4aϵr)2 + 8r
√a2 − 4aϵr(a+
√a2 − 4aϵr)
](6.39)
In this case system is linear stable if a ≥ 4ϵr.
In the case of extinction, i.e. x03 = 0 = y03, (6.36) becomes
J =
r 0
0 −c
. (6.40)
In this case, since N = −12(r − c) and Z = −rc, the deviation curvature for the
case of extinction is
P =1
4(r + c)2. (6.41)
For the Jacobi stable, P should be negative or zero, otherwise Jacobi unstable but
here P is always positive. Therefore the extinction state is the Jacobi unstable
where as system is linear stable if N > 0, i. e., if r < c, and linear unstable if
r > c.
6.5 Numerical simulations and discussion
6.5.1 Table
78
Cases Linear Stability Jacobi Stability
Fixed no. of prey Co-existence Always linear Jacobi stable if
using refuges of species stable b2m2r ≤ 4c2(bm+ c)
Extinction Linear stable if Always Jacobi
of species bm+ c > r unstable
Constant proportion of Co-existence bifurcation Always Jacobi
prey using refuges of species stable
Extinction Linear stable if Always Jacobi
of species r < c unstable
Function of predator-prey Co-existence Linear stable if Jacobi unstable
encounter using refuges of species a ≥ 4ϵr if a ≥ 4ϵr
Extinction Linear stable if Always Jacobi
of species r < c unstable
From table (6.5.1), we notice the following observations:
(i) For co-existence of species, when fixed number of prey using refuges, it is
observed that system is always linearly stable whereas Jacobi stability if system
depends on the prey refuge parameter m. Inequality b2m2r ≤ 4c2(bm + c), may
not be satisfied for large values of m. This reflects the ecological meaning of
predator-prey system that if more prey population will use refuge, then predator
population will go to extinct.
(ii) Again, for co-existence of species, when constant proportion of prey using
refuges, it is observed that in case of linear stability bifurcation occurs, whereas
system is always Jacobi stable.
(iii) Lastly, when a function of predator-prey encounter using refuges, system
is linearly stable if a > 4ϵr, whereas under the same condition system is Jacobi
unstable. From a > 4ϵr, we note that this inequality may be satisfied for larger
values of ϵ, i. e. if predator-prey encounters are occur more frequently then prey
population will go towards extinction.
79
From all these observations, we can say that, study of Jacobi stability behavior
of systems reflects the better ecological interpretation.
To substantiate our all analytical findings numerically, we consider the fol-
lowing set of parameter values r = 10, c = 3, a = 1.5 and b = 0.8.
Values of parameters are hypothetical and do not necessarily have a biological
meaning.
Figures 1-8, are the plots of prey and predator population versus time for
different values of prey refuge parameter m(= 0.2, 0.5, 0.8, and 3.0) when constant
numbers of prey are using refuges. From these figures, we observe that oscil-
latory behavior of these populations decreases with increase in value of m and
consequently populations attain their equilibrium levels in less time.
Figures 09-14, are the plots of prey and predator population versus time
for different values of η(= 0.2, 0.5, and 0.8) when constant proportion of prey
using refuges. From these figures, it can be depicted that both population never
attain their equilibrium level, because oscillatory behavior and also peak value of
oscillations increases with time. This is the case of bifurcation.
Figures 15-18, are the plots of prey and predator populations against time
for different values of ϵ(= 0.02 and 0.03) prey refuge parameter when function of
prey and predator encounters using refuges. These figures show that, equilibrium
values of both populations increase with ϵ. It is also observed that oscillatory
behavior decreases with time and ultimately populations attain their equilibrium
level.
Figures 19-22, are the phase planes of prey versus predator population for
different values of m(= 0.2, 0.5, 0.8, and 3.0). For these values of m, we obtain
stable spirals showing that populations eventually attain their equilibrium level.
This is the case of global stability for any system.
Figures 23-25, are the phase planes of prey versus predator population for
different values of η(= 0.2, 0.5, and 0.8). Here, we note that bifurcation (neither
stable nor unstable spirals) occurs for each value of η.
Figures 26-27, are the phase planes of prey versus predator population for
different values of ϵ(= 0.02 and 0.03). Here, we obtain stable spirals for these
80
0 5 10 15 201
2
3
4
5
6
7
8
Time (t)
prey
(x)
m=0.2
Fig. 1: Time vs prey for m=0.2
0 5 10 15 202
3
4
5
6
7
8
9
Time (t)
prey
(x)
m=0.5
Fig. 2: Time vs prey for m=0.5
0 5 10 15 202
3
4
5
6
7
8
9
10
Time (t)
prey
(x)
m=0.8
Fig. 3: Time vs prey for m=0.8
0 5 10 15 202
4
6
8
10
12
14
16
Time (t)
prey
(x)
m=3
Fig. 4: Time vs prey for m=3
0 5 10 15 204
5
6
7
8
9
10
11
Time (t)
pred
ator
(y)
m=0.2
Fig. 5: Time vs predator for m=0.2
0 5 10 15 204
5
6
7
8
9
10
11
12
Time (t)
pred
ator
(y)
m=0.5
Fig. 6: Time vs predator for m=0.5
81
0 5 10 15 204
5
6
7
8
9
10
11
12
Time (t)
pred
ator
(y)
m=0.8
Fig. 7: Time vs predator for m=0.8
0 5 10 15 202
4
6
8
10
12
14
16
18
Time (t)
pred
ator
(y)
m=3
Fig. 8: Time vs predator for m=3
0 5 10 15 200
2
4
6
8
10
12
14
Time (t)
prey
(x)
η=0.2
Fig. 9: Time vs prey for η = 0.2
0 5 10 15 200
5
10
15
20
25
30
35
Time (t)
prey
(x)
η=0.5
Fig. 10: Time vs prey for η = 0.5
0 5 10 15 200
50
100
150
Time (t)
prey
(x)
η=0.8
Fig. 11: Time vs prey for η = 0.8
0 5 10 15 202
4
6
8
10
12
14
16
Time (t)
pred
ator
(y)
η=0.2
Fig. 12: Time vs predator for η = 0.2
82
0 5 10 15 200
5
10
15
20
25
30
35
Time (t)
pred
ator
(y)
η=0.5
Fig. 13: Time vs predator for η = 0.5
0 5 10 15 200
20
40
60
80
100
120
140
Time (t)
pred
ator
(y)
η=0.8
Fig. 14: Time vs predator for η = 0.8
0 5 10 15 202
3
4
5
6
7
8
9
10
Time (t)
prey
(x)
ε=0.02
Fig. 15: Time vs prey for ϵ = 0.02
0 5 10 15 202
4
6
8
10
12
Time (t)
prey
(x)
ε=0.03
Fig. 16: Time vs prey for ϵ = 0.03
0 5 10 15 204
5
6
7
8
9
10
11
12
13
Time (t)
pred
ator
(y)
ε=0.02
Fig. 17: Time vs predator for ϵ = 0.02
0 5 10 15 204
6
8
10
12
14
16
Time (t)
pred
ator
(y)
ε=0.03
Fig. 18: Time vs predator for ϵ = 0.03
83
0 2 4 6 84
5
6
7
8
9
10
11
prey (x)
pred
ator
(y)
m=0.2
Fig. 19: Prey vs predator for m=0.2
2 4 6 8 104
5
6
7
8
9
10
11
12
prey (x)
pred
ator
(y)
m=0.5
Fig. 20: Prey vs predator for m=0.5
2 4 6 8 104
5
6
7
8
9
10
11
12
prey (x)
pred
ator
(y)
m=0.8
Fig. 21: Prey vs predator for m=0.8
0 5 10 15 202
4
6
8
10
12
14
16
18
prey (x)
pred
ator
(y)
m=3
Fig. 22: Prey vs predator for m=3
0 5 10 152
4
6
8
10
12
14
16
prey (x)
pred
ator
(y)
η=0.2
Fig. 23: Prey vs predator for η = 0.2
0 5 10 15 20 25 30 350
5
10
15
20
25
30
35
prey (x)
pred
ator
(y)
η=0.5
Fig. 24: Prey vs predator for η = 0.5
84
0 50 100 1500
20
40
60
80
100
120
140
prey (x)
pred
ator
(y)
η=0.8
Fig. 25: Prey vs predator for η = 0.8
2 4 6 8 104
6
8
10
12
14
prey (x)
pred
ator
(y)
ε=0.02
Fig. 26: Prey vs predator for ϵ = 0.02
2 4 6 8 10 124
6
8
10
12
14
16
prey (x)
pred
ator
(y)
ε=0.03
Fig. 27: Prey vs predator for ϵ = 0.03
85
values of ϵ.
6.6 Conclusions
In this chapter, we have proposed and analyzed predator-prey models in three
cases: 1) when constant number of preys are using refuges, 2) when constant
proportion of preys are using refuges and 3) when a function of predator-prey
encounter using refuges. Analyses of these models have been done using the Lya-
punov and KCC theory of stability. Jacobi stability behaviors have been studied
by obtaining the deviation curvature in each case. Stability results have been com-
pared for both cases. In case when constant proportion of prey is using refuges, we
have observed that Jacobi stability and linear stability behaviors of populations
completely differ from each other. In the case when a function of prey-predator
encounters using refuges, it has been observed that, inequality under which the
system is linearly stable also holds for Jacobi instability of the system. From all
these findings, we have also pointed out a very important observation that Jacobi
stability analysis reveals the better ecological interpretation of any system. This
is so because Jacobi stability analysis also includes the non-linear terms of differ-
ential equations but in linear stability analysis we omit non-linear terms.
The models presented in this chapter are of great use to maintain the stability of
parks where prey and predator population live together.
86
Chapter 7Finsler-Randers Cosmological modelsin Modified Gravity Theories
In this chapter, Finsler-Randers cosmological models in modified gravity theories
have been investigated. The de Sitter, power law and general exponential solu-
tions are assumed for the scale factor in the corresponding cosmological models.
For each scenario, we have discussed all energy conditions in detail. We have
also investigated the behaviour of FR cosmological models in modified theories of
gravity like Einstein theory, Hoyle-Narlikar Creation field theory, Lyra geometry
and General class of scalar-tensor theories.
7.1 Introduction
Particular attention over the last decade has been paid on the so-called Finsler-
Randers (hereafter FR) cosmological model [Randers (1941)]. In general metrical
extensions of Riemann geometry can provide a Finslerian geometrical structure
in a manifold which leads to generalized gravitational field theories. During the
last decade there is a rapid development of applications of Finsler geometry in its
FR context, mainly in the topics of general relativity, astrophysics and cosmology.
The spatially homogeneous cosmological models allow extension of cosmological
investigations to distorting and rotating universes, giving estimates of effects of
anisotropy on primordial element production and on the measured CMBR spec-
trum anisotropy [Ellis and Elst (1999)]. Apart from observational reasons, there
are various theoretical considerations that have motivated the study of anisotropic
cosmologies. Among these are (i) some kind of singularity in our “past" is strongly
87
indicated if certain reasonable conditions hold [Hawking and Ellis (1973)]. How-
ever, it could differ greatly from the type found in FRW models [Belinski et al.
(1970)]. (ii) The “Chaotic Cosmology” programme of Misner (1968) sought mech-
anism to explain why the observed isotropy and homogeneity should exist regard-
less of the initial conditions [MacCallum (1979), Ellis (1993) and Kolb and Turner
(1990)].
There exist wide class of anisotropic cosmological models, which are often
studied in cosmology [Misner et al. (1973)]. There are theoretical arguments that
sustain the existence of an anisotropic phase that approaches an isotropic case
[Misner (1968)] (Chaotic Cosmology). Also, anisotropic cosmological models are
found a suitable candidate to avoid the assumption of specific initial conditions
in FRW models. The early Universe could also be characterized by irregular ex-
pansion mechanism. Therefore, it would be useful to explore cosmological models
in which anisotropies, existing at early stage of expansion, are damped out in
the course of evolution. Interest in such models have received attention [Hu and
Parker (1978)].
Stavrinos (2008) have studied the Friedman-like Robertson-Walker model in
generalized metric space time with weak anisotropy. Recently, Basilakos and
Stavrinos (2013) have studied cosmological equivalence between the Finsler-Randers
space time and the DGP gravity model. This motivates the researchers to con-
sider the model of Universe with Finsler-Randers space time cosmology. Given
the growing interest of cosmologists, here, we propose to study the evolution of
the Universe within the framework of Finsler-Randers cosmology. In this chapter
we have studied the Friedman Robertson Walker model in the Finsler-Randers
cosmology. The out line of the chapter is as follows: In section 1, a brief introduc-
tion is given. In section 2, the metric and the basic field equations are described
in Finsler-Randers cosmology. Section 3 deals with the FR cosmological model in
Lyra geometry. FR cosmological model in general class of scalar tensor theories
and C-field theory are given in section 4 and 5, respectively. Finally, conclusions
are summarized in the last section 6.
88
7.2 Finsler-Randers Cosmological Model in Ein-
stein Theory
The FR cosmic scenario is based on the Finslerian geometry which extends the
Riemannian geometry. Notice that a Riemannian geometry is also a particular
case of Finslerian. Bellow we discuss only the main features of the theory [for
more details see Rund (1959), Miron and Anastasiei (1994), Bao et al. (2000)
and Vacaru et al. (2005)]. Generally, a Finsler space is derived from a generating
function F (x, y) on the tangent bundle TM of a manifold M . The generating
function F is differentiable on TM0 = TM \ 0 and continuous on the zero
cross section. The function F is also positively homogeneous of degree-one in
y = x = dxdt
. In other words, F introduces a structure on the space-time manifold
M that is called Finsler space-time. In the case of a FR space-time we have
F (x, y) =√aij(x)yiyj + bi(x)y
i, (7.1)
where aij are component of a Riemannian metric and bi = (b0, 0, 0, 0) is a weak
primordial vector field with |bi| << 1. Now the Finslerian metric tensor gij is
constructed by the Hessian of F 2
2
gij =1
2
∂2F 2
∂yi∂yj. (7.2)
It is interesting to mention that the Cartan tensor Cijk = 12
∂gij∂yk
= 14
∂3F 2
∂yi∂yj∂yk
is a significant ingredient of the Finsler geometry. Indeed it has been found by
Stavrinos (2008) that b0 = 2C000.
The Finslerian-Randers field equations are given by
Rij −1
2gijR = −8πG
c4Tij, (7.3)
where Rij is the Finslerian Ricci tensor, Tij is the energy-momentum tensor
and T is the trace of the energy-momentum tensor. Modelling the expanding
Universe as a Finslerian perfect fluid with four-velocity Ui for co-moving observers,
we have
89
Tij = −pgij + (ρ+ p)UiUj, (7.4)
where ρ and p are the total energy density and pressure of the cosmic fluid
respectively.
Thus the energy momentum tensor becomes
Tij = diag(ρ,−pg11,−pg22,−pg33). (7.5)
Following the work of Kolassis et al. (1988), Chatterjee and Banerjee (2004) and
Bali and Saraf (2013), we discuss briefly week, dominant and strong energy con-
ditions in the context of Finslerian cosmology for our model.
We have T 00 = ρ, T 1
1 = T 22 = T 3
3 = −p in the locally Minkowskian frame.
Obviously the roots of matrix equation
|Tij − rgij| = dia[(ρ− r), (r + p), (r + p), (r + p)] = 0, (7.6)
give the eigenvalues r for our energy momentum tensor as r0 = ρ and r1 = −p =
r2 = r3. The energy conditions for our model are given by :
• Null Energy Condition (NEC):
ρ+ p ≥ 0. (7.7)
• Weak Energy Condition (WEC):
r0 ≥ 0 ⇒ ρ ≥ 0; r0 − ri ≥ 0 ⇒ ρ+ p ≥ 0. (7.8)
• Strong Energy Condition (SEC):
r0 −∑
ri ≥ 0 ⇒ ρ+ 3p ≥ 0 and ρ+ p ≥ 0. (7.9)
• Dominant Energy Condition (DEC):
90
r0 ≥ 0 ⇒ ρ ≥ 0; −r0 ≤ −ri ≤ r0 ⇒ ρ± p ≥ 0. (7.10)
In the context of a FRW metric
aij = diag
(1,− a2
1− kr2,−a2r2,−a2r2sin2θ
), (7.11)
where a is the function of time t only and k is the curvature parameter
having values +1, 0,−1 for closed, flat and open models respectively. The non-
zero components of the Finslerian Ricci tensors are
R00 = 3
(a
a− 3
4
a
au0
)(7.12)
and
Rii = −
(aa+ 2a2 + 2k + 11
4aau0
ii
), (7.13)
where 11 = 1− kr2, 22 = r2 and 33 = r2sin2θ.
The gravitational FR field equations (7.1), for co-moving observers, FRW Einstein
Field equations are
a
a+
3
4
a
aZt = −4πG
3(ρ+ 3p) (7.14)
a
a+ 2
(a
a
)2
+ 2k
a2+
11
4
a
aZt = 4πG (ρ− p) , (7.15)
where the over-dot denotes derivative with respect to the cosmic time ‘t’ and
Zt = b0 < 0 [Stavrinos (2008)].
From equations (7.2) and (7.3), we get
H2 +k
a2+HZt =
8πG
3ρ. (7.16)
Obviously, the extra term H(t)Zt in the modified Friedmann equation (7.16)
affects the dynamics of the Universe. If we consider b0 ≡ 0 or (C000 ≡ 0, Fα= 1),
which implies Zt = 0, then the field equations (7.14) and (7.15) reduce to the
nominal Einstein’s equations, a solution of which is the usual Friedman equation.
91
Here we discuss two different physically viable cosmologies, which have physical
interests to describe the decelerating and accelerating phases of universe.
Case 1: de Sitter solution
de Sitter solutions are well known in Cosmology because the current epoch,
wherein the Universe expansion is being accelerated, can be described approxi-
mately with a de Sitter solutions. This kind of solution consists of an exponential
expansion of the scale factor, which yields a constant Hubble parameter.
We consider the following form for a scale factor by Arbab. (2007): a = ceγt, where
c and γ are constants. For γ2 > 0 it gives an accelerating Universe.
Now, Hubble parameter
H =a
a=γceγt
ceγt= γ. (7.17)
Using equations (7.4) and (7.5), the energy density evolving as
ρ =3
8πG
[γ2 + γZt +
k
c2e2γt
]. (7.18)
From equations (7.2) and (7.6), the pressure is given by
p = − 3γ2
8πG− 5
16πGγZt −
k
8πGc2e2γt. (7.19)
Using equations (7.6) and (7.7), we obtain
ρ+ p =γZt
16πG+
1
4πG
k
c2e2γt. (7.20)
and
ρ− p =3γ2
4πG+
11γZt
16πG+
1
2πG
k
c2e2γt. (7.21)
Again, from equations (7.6) and (7.7), one can get
ρ+ 3p = − 3γ2
4πG− 9
16πGγZt. (7.22)
It is noticed that, the equation (7.16) reduces to standard Friedmann equa-
tion by considering Zt = 0. The current observations says that the Universe
92
is anisotropic at early stage and becomes isotropic at late time (i.e. at present
epoch). So keeping these things in mind, we consider the physically variable
choices of Zt < 0 in two different scenario. i.e. (i) Zt = −e−t and (ii) Zt = −t−n.
Subcase 1a: When Zt = −e−t
Substituting the value of Zt in equations (7.18) and (7.19), we can determine
ρ and p respectively as
ρ =3
8πG
[γ2 − γe−t +
k
c2e2γt
]. (7.23)
p =5γe−t
16πG− 3γ2
8πG− k
8πGc2e2γt. (7.24)
From equations (7.23) and (7.24), we obtain
ρ+ p = − γe−t
16πG+
1
4πG
k
c2e2γt. (7.25)
and
ρ− p =3γ2
4πG− 11γe−t
16πG+
1
2πG
k
c2e2γt. (7.26)
Again, from equations (7.23) and (7.24), one can obtain easily
ρ+ 3p = − 3γ2
4πG+
9γe−t
16πG. (7.27)
From equations (7.23)-(7.27), it is observed that the NEC is satisfied if
c2 ≤ 4kγe(2γ+1)t = A1, WEC is satisfied for c2 ≤ min
k
e2γt[γe−t−γ2], 4kγe(2γ+1)t
= A2,
DEC is satisfied if c2 ≤ min
ke2γt[γe−t−γ2]
, 4kγe(2γ+1)t ,
8ke2γt[11γe−t−12γ2]
= A3 and SEC
is satisfied if 0 < γ ≤ 34et
.
It is observed that, for any value of t, NEC, WEC and DEC are satisfied in this
case if c2 ≤ min A1, A2, A3 whereas SEC is satisfied in this model if 0 < γ ≤ 34et
.
However, we also observed that for large cosmic time ‘t’, NEC, WEC and DEC
are satisfied whereas SEC is violated, which is responsible for current accelerated
93
expansion of Universe.
Subcase 1b: When Zt = −t−n
Substituting the value of Zt in equations (7.18) and (7.19), we can determine
ρ and p respectively as
ρ =3
8πG
[γ2 − γ
tn+
k
c2e2γt
](7.28)
and
p =5γ
16πGtn− 3γ2
8πG− k
8πGc2e2γt. (7.29)
From equations (7.28) and (7.29), we obtain
ρ+ p = − γt−n
16πG+
1
4πG
k
c2e2γt(7.30)
and
ρ− p =3γ2
4πG− 11γt−n
16πG+
1
2πG
k
c2e2γt. (7.31)
Again, from equations (7.28) and (7.29), we can obtain easily
ρ+ 3p =9γt−n
16πG− 3γ2
4πG. (7.32)
From equations (7.28)-(7.32), it is observed that the NEC is satisfied if
c2 ≤ 4kγe2γttn
= A4, WEC is satisfied for c2 ≤ min
ke2γt[γt−n−γ2]
, 4kγtne(2γ)t
= A5,
DEC is satisfied if c2 ≤ min
ke2γt[γt−n−γ2]
, 4kγtne(2γ)t
, 8ke2γt[11γt−n−12γ2]
= A6 and SEC
is satisfied if 0 < γ ≤ 34tn
.
It is observed that, for any value of t, NEC, WEC and DEC are satisfied in this
case if c2 ≤ min A4, A5, A6 whereas SEC is satisfied in this model if 0 < γ ≤ 34tn
.
However, we also observed that for large cosmic time ‘t’, NEC, WEC and DEC
are satisfied but SEC is violated.
94
Case 2: Power-law solution
We know that the power law solutions are very important in the standard
Cosmology, because this type of solution provides a framework for establishing
the behaviour of more general cosmological solutions in different histories of our
Universe, such as radiation-dominant, matter-dominant, or dark energy-dominant
eras. Let us consider a Universe with a power law by Padmanabhan (2002):
a = ctδ, where c and δ are constants. For δ > 1 it gives an accelerating Universe.
Now, Hubble parameter
H =a
a=cδtδ−1
ctδ=δ
t. (7.33)
Using equations (7.16) and (7.33), the energy density evolving as
ρ =3
8πG
[δ2
t2+δ
tZt +
k
c2t2δ
]. (7.34)
From equations (7.14) and (7.34), the pressure is given by
p =−3δ2 + 2δ
8πGt2− 5δZt
16πGt− 1
8πG
k
c2t2δ. (7.35)
From equations (7.34) and (7.35), we obtain
ρ+ p =1
4πG
δ
t2+
1
16πG
δ
tZt +
1
4πG
k
c2t2δ(7.36)
and
ρ− p =1
4πG
δ(3δ − 1)
t2+
11
16πG
δ
tZt +
1
2πG
k
c2t2δ. (7.37)
Again, from equations (7.34) and (7.35), we can obtain easily
ρ+ 3p =3
4πG
δ(1− δ)
t2− 9
16πG
δ
tZt. (7.38)
Subcase 2a: When Zt = −e−t
Substituting the value of Zt in equations (7.34) and (7.35), we can determine
ρ and p respectively as
95
ρ =3
8πG
[δ2
t2− δ
tet+
k
c2t2δ
], (7.39)
p =2δ − 3δ2
8πGt2+
5δ
16πGtet− 1
8πG
k
c2t2δ. (7.40)
From equations (7.39) and (7.40), we obtain
ρ+ p =1
4πG
δ
t2− 1
16πG
δ
tet+
1
4πG
k
c2t2δ(7.41)
and
ρ− p =δ(3δ − 1)
4πGt2− 11
16πG
δ
tet+
1
2πG
k
c2t2δ. (7.42)
Again, from equations (7.39) and (7.40), we obtain ρ+ 3p, where
ρ+ 3p =3δ(1− δ)
4πGt2+
9
16πG
δ
tet. (7.43)
From equations (7.39)-(7.43), it is observed that the NEC is satisfied if
c2 ≤ 4kδt(2δ−1)e−t−4δ2t(2δ−2) = B1, WEC is satisfied for
c2 ≤ min
kδt2δ−1e−t−δ2t2δ−2 ,
4kδt(2δ−1)e−t−4δ2t(2δ−2)
= B2, DEC satisfied if
c2 ≤ min
kδt2δ−1e−t−δ2t2δ−2 ,
4kδt(2δ−1)e−t−4δ2t(2δ−2) ,
8k11δt2δ−1e−t−4δ(3δ−1)t2δ−2
= B3 and
SEC satisfied if 1 < δ ≤ 1 + 3t4et
. It is observed that, for any value of ‘t’, NEC,
WEC and DEC are satisfied in this case if c2 ≤ min B1, B2, B3 whereas SEC is
satified in this model if 1 < δ ≤ 1 + 3t4et
. It is also observed that for large cosmic
time ‘t’, NEC, WEC, DEC are stasfied but SEC is violated.
Subcase 2b: When Zt = −t−n
Substituting the value of Zt in equations (7.34) and (7.35), we can determine
ρ and p respectively as
ρ =3
8πG
[δ2
t2− δ
tn+1+
k
c2t2δ
], (7.44)
96
p =2δ − 3δ2
8πGt2+
5δ
16πGtn+1− 1
8πG
k
c2t2δ. (7.45)
From equations (7.44) and (7.45), we obtain
ρ+ p =1
4πG
δ
t2− 1
16πG
δ
tn+1+
1
4πG
k
c2t2δ(7.46)
and
ρ− p =δ(3δ − 1)
4πGt2− 11
16πG
δ
tn+1+
1
2πG
k
c2t2δ. (7.47)
Again, from equations (7.44) and (7.45), we can obtain easily
ρ+ 3p =3δ(1− δ)
4πGt2+
9
16πG
δ
tn+1. (7.48)
From equations (7.44)-(7.48), it is observed that the NEC is satisfied if
c2 ≤ 4kδt2δ−n−1−δt2δ−2 = B4, WEC is satisfied for
c2 ≤ min
kδt2δ−n−1−δ2t2δ−2 ,
4kδt2δ−n−1−δt2δ−2
= B5, DEC is satisfied if
c2 ≤ min
kδt2δ−n−1−δ2t2δ−2 ,
4kδt2δ−n−1−δt2δ−2 ,
8k11δt2δ−n−1−4δ(3δ−1)t2δ−2
= B6 and SEC is
satisfied if 1 < δ ≤ 1 + 34tn−1 .
It is observed that, for any value of ‘t’, NEC, WEC and DEC are satisfied in
this case if c2 ≤ min B4, B5, B6 whereas SEC is satisfied in this model for
1 < δ ≤ 1 + 34tn−1 . It is also observed that for large cosmic time ‘t’, NEC, WEC,
DEC are stasfied but SEC is violated.
7.3 Finsler-Randers cosmological model in Lyra ge-
ometry
The field equation in this theory are given by
Rij −1
2Rgij +
3
2ϕiϕj −
3
4gijϕkϕ
k = −Tij, (7.49)
where ϕij are vector displacement field. Here we take
97
ϕi = (0, 0, 0, β(t)) .
In this theory Field equations are
(a
a
)2
+k
a2+HZt =
8πG
3ρ+
1
2β2, (7.50)
2a
a+
(a
a
)2
+k
a2+
5
2HZt = 8πGp− 1
2β2. (7.51)
From equations (7.50) and (7.51), we obtain
a
a+
3
4
a
aZt = −4πG
3(ρ+ 3p)− 1
2β2. (7.52)
Case 1: de Sitter solution
Let a = ceγt, where γ2 > 0.
Now, Hubble parameter
H =a
a=γceγt
ceγt= γ. (7.53)
From equations (7.52) and (7.53), the energy density evolving as
ρ =3
8πG
[γ2 + γZt +
k
c2e2γt− 1
2β2
]. (7.54)
From equations (7.52) and (7.54), the pressure is given as
p = − 3γ2
8πG− 5
16πGγZt −
k
8πGc2e2γt− 1
16πGβ2. (7.55)
From equations (7.54) and (7.55), we obtain
ρ+ p =γZt
16πG+
1
4πG
k
c2e2γt− 1
4πGβ2 (7.56)
and
ρ− p =3γ2
4πG+
11γZt
16πG+
1
2πG
k
c2e2γt− 1
8πGβ2. (7.57)
Again, from equations (7.54) and (7.55), we can obtain easily
98
ρ+ 3p = − 3γ2
4πG− 9
16πGγZt −
3
8πGβ2. (7.58)
In this section, we discuss two different cases of physically viable cosmologies.
Subcase 1a: When Zt = −e−t :
Substituting the value of Zt in equations (7.54) and (7.55), we can determine
ρ and p respectively as
ρ =3
8πG
[γ2 − γe−t +
k
c2e2γt− 1
2β2
], (7.59)
p =5γe−t
16πG− 3γ2
8πG− k
8πGc2e2γt− 1
16πGβ2. (7.60)
From equations (7.59) and (7.60), we obtain
ρ+ p = − γe−t
16πG+
1
4πG
k
c2e2γt− 1
4πGβ2 (7.61)
and
ρ− p =3γ2
4πG− 11γe−t
16πG+
1
2πG
k
c2e2γt− 1
8πGβ2. (7.62)
Again, from equations (7.59) and (7.60), we obtain easily
ρ+ 3p = − 3γ2
4πG+
9γe−t
16πG− 3
8πGβ2. (7.63)
From equations (7.59)-(7.63), it is noticed that the NEC is satisfied if
c2 ≤ 4ke2γt[γe−t−4β2]
= A1, WEC is satisfied for c2 ≤ min
ke2γt[γe−t−γ2+β2/2]
, 4ke2γt[γe−t−4β2]
=
A2 DEC is satisfied if c2 ≤ min
ke2γt[γe−t−γ2+β2/2]
, 4ke2γt[γe−t−4β2]
, 8ke2γt[11γe−t−12γ2−2β2]
=
A3 and SEC is satisfied if 0 < β2 ≤ 3γ2et
− 2γ2.
It is observed that, for any value of t, NEC, WEC and DEC are satisfied in
this case if c2 ≤ min A1, A2, A3 whereas SEC is satisfied in this model for
0 < β2 ≤ 3γ2et
− 2γ2. However, we also observed that for large cosmic time ‘t’,
NEC, WEC and DEC are satisfied but SEC is violated.
99
Subcase 1b: When Zt = −t−n :
Substituting the value of Zt in equations (7.54) and (7.55), we can determine
ρ and p respectively as
ρ =3
8πG
[γ2 − γ
tn+
k
c2e2γt− 1
2β2
], (7.64)
p =5γ
16πGtn− 3γ2
8πG− k
8πGc2e2γt− 1
16πGβ2. (7.65)
From equations (7.64) and (7.65), we obtain
ρ+ p = − γt−n
16πG+
1
4πG
k
c2e2γt− 1
4πGβ2 (7.66)
and
ρ− p =3γ2
4πG− 11γt−n
16πG+
1
2πG
k
c2e2γt− 1
8πGβ2. (7.67)
Again, from equations (7.64) and (7.65), we obtain
ρ+ 3p =9γt−n
16πG− 3γ2
4πG− 3
8πGβ2. (7.68)
From equations (7.64)-(7.68), it is noticed that the NEC is satisfied if
c2 ≤ 4ke2γt[γt−n+4β2]
= A4, WEC is satisfied for c2 ≤ min
ke2γt[γt−n−γ2+β2/2]
, 4ke2γt[γt−n+4β2]
=
A5, DEC is satisfied if c2 ≤ min
ke2γt[γt−n−γ2+β2/2]
, 4ke2γt[γt−n+4β2]
, 8ke2γt[11γt−n−12γ2+2β2]
=
A6 and SEC is satisfied if 0 < β2 ≤ 3γ2tn
− 2γ2
It is observed that, for any valu of t, NEC, WEC and DEC are satisfied in
this case if c2 ≤ min A4, A5, A6 whereas SEC is satisfied in this model if
0 < β2 ≤ 3γ2tn
− 2γ2. However, we also observed that for large cosmic time ‘t’,
NEC, WEC and DEC are satisfied but SEC is violated.
Case 2: Power-law
Let a = ctδ, where δ > 1.
Now, Hubble parameter
100
H =a
a=cδtδ−1
ctδ=δ
t. (7.69)
From equations (7.50) and (7.69), we have
ρ =3
8πG
[δ2
t2+δ
tZt +
k
c2t2δ− 1
2β2
]. (7.70)
From equations (7.50) and (7.70), we get
p =−3δ2 + 2δ
8πGt2− 5δZt
16πGt− 1
8πG
k
c2t2δ− 1
16πGβ2. (7.71)
From equations (7.70) and (7.71), we obtain
ρ+ p =1
4πG
δ
t2+
1
16πG
δ
tZt +
1
4πG
k
c2t2δ− 1
4πGβ2 (7.72)
and
ρ− p =1
4πG
δ(3δ − 1)
t2+
11
16πG
δ
tZt +
1
2πG
k
c2t2δ− 1
8πGβ2. (7.73)
Again, from equations (7.70) and (7.71), we obtain
ρ+ 3p =3
4πG
δ(1− δ)
t2− 9
16πG
δ
tZt −
3
8πGβ2. (7.74)
Subcase 2a: When Zt = −e−t :
Substituting the value of Zt in equations (7.70) and (7.71), we can determine
ρ and p respectively as
ρ =3
8πG
[δ2
t2− δ
tet+
k
c2t2δ− 1
2β2
], (7.75)
p =2δ − 3δ2
8πGt2+
5δ
16πGtet− 1
8πG
k
c2t2δ− 1
16πGβ2. (7.76)
From equations (7.75) and (7.76), we obtain
ρ+ p =1
4πG
δ
t2− 1
16πG
δ
tet+
1
4πG
k
c2t2δ− 1
4πGβ2 (7.77)
101
and
ρ− p =δ(3δ − 1)
4πGt2− 11
16πG
δ
tet+
1
2πG
k
c2t2δ− 1
8πGβ2. (7.78)
Again, from equations (7.75) and (7.76), we obtain
ρ+ 3p =3δ(1− δ)
4πGt2+
9
16πG
δ
tet− 3
8πGβ2. (7.79)
From equations (7.75)-(7.79), it is observed that the NEC is satisfied if
c2 ≤ 4kδt(2δ−1)e−t−4δ2t(2δ−2)+4β2t2δ
= B1, WEC is satisfied for
c2 ≤ min
kδt2δ−1e−t−δ2t2δ−2+t2δβ2/2
, 4kδt(2δ−1)e−t−4δ2t(2δ−2)+4β2t2δ
= B2,
DEC is satisfied if
c2 ≤ min
kδt2δ−1e−t−δ2t2δ−2+t2δβ2/2
, 4kδt(2δ−1)e−t−4δ2t(2δ−2)+4β2t2δ
, 8k11δt2δ−1e−t−4δ(3δ−1)t2δ−2+2β2t2δ
=
B3 and SEC is satisfied if 0 < β2 ≤ 2δ(1−δ)t2
+ 3δ2tet
.
It is observed that, for any value of t, NEC, WEC and DEC are satisfied in
this case if c2 ≤ min B1, B2, B3 whereas SEC is satisfied in this model if
0 < β2 ≤ 2δ(1−δ)t2
+ 3δ2tet
. However, we also observed that for large cosmic time
‘t’, NEC, WEC and DEC are satisfied but SEC is violated.
Subcase 2b: WhenZt = −t−n
Substituting the value of Zt in equations (7.70) and (7.71), we can determine
ρ and p respectively as
ρ =3
8πG
[δ2
t2− δ
tn+1+
k
c2t2δ− 1
2β2
], (7.80)
p =2δ − 3δ2
8πGt2+
5δ
16πGtn+1− 1
8πG
k
c2t2δ− 1
16πGβ2. (7.81)
From equations (7.80) and (7.81), we obtain
ρ+ p =1
4πG
δ
t2− 1
16πG
δ
tn+1+
1
4πG
k
c2t2δ− 1
4πGβ2 (7.82)
and
102
ρ− p =δ(3δ − 1)
4πGt2− 11
16πG
δ
tn+1+
1
2πG
k
c2t2δ− 1
8πGβ2. (7.83)
Again, from equations (7.80) and (7.81), we obtain
ρ+ 3p =3δ(1− δ)
4πGt2+
9
16πG
δ
tn+1− 3
8πGβ2. (7.84)
From equations (7.80)-(7.84), it is observed that the NEC is satisfied if
c2 ≤ 4kδt2δ−n−1−δt2δ−2+t2δβ2/2
= B4, WEC is satisfied for
c2 ≤ min
kδt2δ−n−1−δ2t2δ−2+4t2δβ2 ,
4kδt2δ−n−1−δt2δ−2+t2δβ2/2
= B5,
DEC satisfied if
c2 ≤ min
kδt2δ−n−1−δ2t2δ−2+4t2δβ2 ,
4kδt2δ−n−1−δt2δ−2+t2δβ2/2
, 8k11δt2δ−n−1−4δ(3δ−1)t2δ−2+2t2δβ2
=
B6 and SEC is satisfied if 0 < β2 ≤ 2δ(1−δ)t2
+ 3δ2tn+1 .
It is observed that, for any value of t, NEC, WEC, and DEC are satisfied in
this case if c2 ≤ min B4, B5, B6 whereas SEC is satisfied in this model if
0 < β2 ≤ 2δ(1−δ)t2
+ 3δ2tn+1 . However, we also observed that for large cosmic time ‘t’,
NEC, WEC and DEC are satisfied but SEC is violated.
7.4 Finsler-Randers Cosmological Model in Gen-
eral Class of Scalar-tensor theory
The field equation of this theory are given by
Gij = Rij −1
2gijR = −8πGTij + 2
(ϕ,iϕ,j −
1
2gijϕ,kϕ
k
), (7.85)
where G is the gravitational constant and ϕ is the scalar field.
In this theory Field equation’s are
3
[H2 +
k
a2+HZt
]= 8πGρ+ ϕ2, (7.86)
2a
a+
(a
a
)2
+k
a2+
5
2HZt = −8πGp− ϕ2. (7.87)
From equations (7.86) and (7.87), we obtain
103
3
[a
a+
3
4
a
aZt
]= −4πG (ρ+ 3p)− 2ϕ2. (7.88)
Case 1: de Sitter solution
Let a = ceγt, where γ2 > 0.
Now, Hubble parameter
H =a
a=γceγt
ceγt= γ. (7.89)
From equations (7.88) and (7.89), we have
ρ =3
8πG
[γ2 + γZt +
k
c2e2γt− ϕ2
3
]. (7.90)
From equations (7.88) and (7.90), we get
p = − 3γ2
8πG− 5
16πGγZt −
k
8πGc2e2γt− 1
8πGϕ2. (7.91)
From equations (7.90) and (7.91), we obtain
ρ+ p =γZt
16πG+
1
4πG
k
c2e2γt− 1
4πGϕ2 (7.92)
and
ρ− p =3γ2
4πG+
11γZt
16πG+
1
2πG
k
c2e2γt. (7.93)
Again, from equations (7.90) and (7.91), we obtain
ρ+ 3p = − 3γ2
4πG− 9
16πGγZt −
1
2πGϕ2. (7.94)
In this section, we discuss two different cases of physically viable cosmologies.
Subcase 1a: When Zt = −e−t
Substituting the value of Zt in equations (7.90) and (7.91), we can determine
ρ and p respectively as
104
ρ =3
8πG
[γ2 − γe−t +
k
c2e2γt− ϕ2
3
], (7.95)
p =5γe−t
16πG− 3γ2
8πG− k
8πGc2e2γt− 1
8πGϕ2. (7.96)
From equations (7.95) and (7.96), we obtain
ρ+ p = − γe−t
16πG+
1
4πG
k
c2e2γt− 1
4πGϕ2 (7.97)
and
ρ− p =3γ2
4πG− 11γe−t
16πG+
1
2πG
k
c2e2γt. (7.98)
Again, from equations (7.95) and (7.96), we obtain
ρ+ 3p = − 3γ2
4πG+
9γe−t
16πG− 1
2πGϕ2. (7.99)
From equations (7.95)-(7.99), it is noticed that the NEC is satisfied if
c2 ≤ 4ke2γt[γe−t+4ϕ2]
= A1, WEC is satisfied for c2 ≤ min
ke2γt[γe−t−γ2+ϕ2/3]
, 4ke2γt[γe−t+4ϕ2]
=
A2, DEC is satisfied if c2 ≤ min
ke2γt[γe−t−γ2+ϕ2/3]
, 4ke2γt[γe−t+4ϕ2]
, 8ke2γt[11γe−t−12γ2]
=
A3 and SEC is satisfied if 0 < ϕ2 ≤ 9γ8et
− 32γ2.
It is observed that, for any value of t, NEC, WEC and DEC are satisfied in this
case if c2 ≤ min A1, A2, A3 whereas SEC is satisfied if 0 < ϕ2 ≤ 9γ8et
− 32γ2.
However, we also observed that for large cosmic time ‘t’, NEC, WEC and DEC
are satisfied but SEC is violated.
Subcase 1b: When Zt = −t−n
Substituting the value of Zt in equations (7.54) and (7.55), we can determine
ρ and p respectively as
ρ =3
8πG
[γ2 − γ
tn+
k
c2e2γt− ϕ2
3
], (7.100)
105
p =5γ
16πGtn− 3γ2
8πG− k
8πGc2e2γt− 1
8πGϕ2. (7.101)
From equations (7.100) and (7.101), we obtain
ρ+ p = − γt−n
16πG+
1
4πG
k
c2e2γt− 1
4πGϕ2 (7.102)
and
ρ− p =3γ2
4πG− 11γt−n
16πG+
1
2πG
k
c2e2γt. (7.103)
Again, from equations (7.100) and (7.101), one can obtain easily
ρ+ 3p =9γt−n
16πG− 3γ2
4πG− 1
2πGϕ2. (7.104)
From equations (7.100)-(7.104), it is observed that the NEC is satisfied if
c2 ≤ 4ke2γt[γt−n+4ϕ2]
= A4, WEC is satisfied for c2 ≤ min
ke2γt[γt−n−γ2+ϕ2/3]
, 4ke2γt[γt−n+4ϕ2]
=
A5, DEC is satisfied if c2 ≤ min
ke2γt[γt−n−γ2+ϕ2/3]
, 4ke2γt[γt−n+4ϕ2]
, 8ke2γt[11γt−n−12γ2]
=
A6 and SEC is satisfied if 0 < ϕ2 ≤ 9γ8tn
− 32γ2.
It is observed that, for any value of t, NEC, WEC and DEC are satisfied in
this case if c2 ≤ min A4, A5, A6 whereas SEC is satisfied in this model if
0 < ϕ2 ≤ 9γ8tn
− 32γ2. However, we also observed that for large cosmic time ‘t’,
NEC, WEC and DEC are satisfied but SEC is violated.
Case 2: Power-law
Let a = ctδ, where δ > 1.
Now, Hubble parameter
H =a
a=cδtδ−1
ctδ=δ
t. (7.105)
From equations (7.88) and (7.105), we have
ρ =3
8πG
[δ2
t2+δ
tZt +
k
c2t2δ− ϕ2
3
]. (7.106)
106
From equations (7.88) and (7.106), we get
p =−3δ2 + 2δ
8πGt2− 5δZt
16πGt− 1
8πG
k
c2t2δ− 1
8πGϕ2. (7.107)
From equations (7.106) and (7.107), we obtain
ρ+ p =1
4πG
δ
t2+
1
16πG
δ
tZt +
1
4πG
k
c2t2δ− 1
4πGϕ2 (7.108)
and
ρ− p =1
4πG
δ(3δ − 1)
t2+
11
16πG
δ
tZt +
1
2πG
k
c2t2δ. (7.109)
Again, from equations (7.106) and (7.107), we can obtain easily
ρ+ 3p =3
4πG
δ(1− δ)
t2− 9
16πG
δ
tZt −
1
2πGϕ2. (7.110)
Subcase 2a: When Zt = −e−t
Substituting the value of Zt in equations (7.105) and (7.106), we can determine
ρ and p respectively as
ρ =3
8πG
[δ2
t2− δ
tet+
k
c2t2δ− ϕ2
3
], (7.111)
p =2δ − 3δ2
8πGt2+
5δ
16πGtet− 1
8πG
k
c2t2δ− 1
8πGϕ2. (7.112)
From equations (7.111) and (7.112), we obtain
ρ+ p =1
4πG
δ
t2− 1
16πG
δ
tet+
1
4πG
k
c2t2δ− 1
4πGϕ2 (7.113)
and
ρ− p =δ(3δ − 1)
4πGt2− 11
16πG
δ
tet+
1
2πG
k
c2t2δ. (7.114)
Again, from equations (7.111) and (7.112), we can obtain easily
ρ+ 3p =3δ(1− δ)
4πGt2+
9
16πG
δ
tet− 1
2πGϕ2. (7.115)
107
From equations (7.111)-(7.115), it is observed that the NEC is satisfied if
c2 ≤ 4kδt(2δ−1)e−t−4δt(2δ−2)+4ϕ2t2δ
= B1, WEC is satisfied for
c2 ≤ min
kδt2δ−1e−t−δ2t2δ−2+t2δϕ2/3
, 4kδt(2δ−1)e−t−4δt(2δ−2)+4ϕ2t2δ
= B2,
DEC satisfied if
c2 ≤ min
kδt2δ−1e−t−δ2t2δ−2+t2δϕ2/3
, 4kδt(2δ−1)e−t−4δt(2δ−2)+4ϕ2t2δ
, 8k11δt2δ−1e−t−4δ(3δ−1)t2δ−2
=
B3 and SEC is satisfied if 0 < ϕ2 ≤ 3δ(1−δ)2t2
+ 9δ8tet
.
It is observed that, for any value of t, NEC, WEC and DEC are satisfied in
this case if c2 ≤ min B1, B2, B3 whereas SEC is satisfied in this model for
0 < ϕ2 ≤ 3δ(1−δ)2t2
+ 9δ8tet
. However, we also observed that for large cosmic time ‘t’,
NEC,WEC and DEC are satisfied but SEC is violated.
Subcase 2b: When Zt = −t−n
Substituting the value of Zt in equations (7.105) and (7.106), we can determine
ρ and p respectively as
ρ =3
8πG
[δ2
t2− δ
tn+1+
k
c2t2δ− ϕ2
3
], (7.116)
p =2δ − 3δ2
8πGt2+
5δ
16πGtn+1− 1
8πG
k
c2t2δ− 1
8πGϕ2. (7.117)
From equations (7.116) and (7.117), we obtain
ρ+ p =1
4πG
δ
t2− 1
16πG
δ
tn+1+
1
4πG
k
c2t2δ− 1
4πGϕ2 (7.118)
and
ρ− p =δ(3δ − 1)
4πGt2− 11
16πG
δ
tn+1+
1
2πG
k
c2t2δ. (7.119)
Again, from equations (7.116) and (7.117), one can obtain easily
ρ+ 3p =3δ(1− δ)
4πGt2+
9
16πG
δ
tn+1− 1
2πGϕ2. (7.120)
From equations (7.116)-(7.120), it is observed that the NEC is satisfied if
c2 ≤ 4kδt2δ−n−1−4δt2δ−2+4t2δϕ2 = B4, WEC is satisfied for
108
c2 ≤ min
kδt2δ−n−1−δ2t2δ−2+t2δϕ2/3
, 4kδt2δ−n−1−4δt2δ−2+4t2δϕ2
= B5, DEC is satisfied
if c2 ≤ min
kδt2δ−n−1−δ2t2δ−2+t2δϕ2/3
, 4kδt2δ−n−1−4δt2δ−2+4t2δϕ2 ,
8k11δt2δ−n−1−4δ(3δ−1)t2δ−2
=
B6 and SEC is satisfied if 0 < ϕ2 ≤ 3δ(1−δ)2t2
+ 9δ8tn+1 .
It is observed that, for any value of t, NEC, WEC and DEC are satisfied in
this case if c2 ≤ min B4, B5, B6 whereas SEC is satisfied in this model for
0 < ϕ2 ≤ 3δ(1−δ)2t2
+ 9δ8tn+1 . However, we also observed that for large cosmic time ‘t’,
NEC, WEC and DEC are satisfied whereas SEC is violated.
7.5 Finsler-Randers Cosmological Model in C-field
theory
The field equations in this theory
Rij −1
2gijR = −8πG
[Tij − f
(cicj −
1
2gijckc
k
)], (7.121)
where c(x, t) = c(t), c = dcdt
and f ≥ 0.
In this theory field equation’s are
(a
a
)2
+k
a2+HZt =
8πG
3
(ρ− 1
2f c2), (7.122)
2a
a+
(a
a
)2
+k
a2+
5
2HZt = −8πG
(p− 1
2f c2). (7.123)
From equations (7.29) and (7.30), we obtain
a
a+
3
4
a
aZt = −4πG
3(ρ+ 3p) +
8πG
3f c2. (7.124)
Case 1: de Sitter solution
Let a = αeγt, where γ2 > 0. Here α and γ are constants.
Now, Hubble parameter
H =a
a=γαeγt
αeγt= γ. (7.125)
109
From equations (7.124) and (7.125), we have
ρ =3
8πG
[γ2 + γZt +
k
α2e2γt
]+
1
2f c2. (7.126)
From equations (7.124) and (7.126), we get
p = − 3γ2
8πG− 5
16πGγZt −
k
8πGα2e2γt− 1
2f c2. (7.127)
From equations (7.126) and (7.127), we obtain
ρ+ p =γZt
16πG+
1
4πG
k
α2e2γt(7.128)
and
ρ− p =3γ2
4πG+
11γZt
16πG+
1
2πG
k
α2e2γt+ f c2. (7.129)
Again, from equations (7.126) and (7.127), we obtain ρ+ 3p, where
ρ+ 3p = − 3γ2
4πG− 9
16πGγZt − f c2. (7.130)
In this section, we discuss two different cases of physically viable cosmologies.
Subcase 1a: When Zt = −e−t
Substituting the value of Zt in equations (7.126) and (7.127), we can determine
ρ and p respectively as
ρ =3
8πG
[γ2 − γe−t +
k
c2e2γt
]+
1
2f c2, (7.131)
p =5γe−t
16πG− 3γ2
8πG− k
8πGc2e2γt− 1
2f c2. (7.132)
From equations (7.131) and (7.132), we obtain
ρ+ p = − γe−t
16πG+
1
4πG
k
c2e2γt(7.133)
and
110
ρ− p =3γ2
4πG− 11γe−t
16πG+
1
2πG
k
c2e2γt+ f c2. (7.134)
Again, from equations (7.131) and (7.132), we obtain
ρ+ 3p = − 3γ2
4πG+
9γe−t
16πG− f c2. (7.135)
From equations (7.131)-(7.135), it is noticed that the NEC is satisfied if
c2 ≤ 4ke2γt[γe−t]
= A1, WEC is satisfied for c2 ≤ min
ke2γt[γe−t−γ2]
, 4ke2γt[γe−t]
= A2,
DEC is satisfied if c2 ≤ min
ke2γt[γe−t−γ2]
, 4ke2γt[γe−t−4β2]
, 8ke2γt[11γe−t−12γ2]
= A3 and
SEC is satisfied if 0 < c2 ≤ 9γ16πGfet
− 3γ2
4πGf.
It is observed that, for any value of t, NEC, WEC and DEC are satisfied in
this case if c2 ≤ min A1, A2, A3 whereas SEC is satisfied in this model for
0 < c2 ≤ 9γ16πGfet
− 3γ2
4πGf. However, we also observed that for large cosmic time ‘t’,
NEC, WEC and DEC are satisfied whereas SEC is violated.
Subcase 1b: When Zt = −t−n
Substituting the value of Zt in equations (7.126) and (7.127), we can determine
ρ and p respectively as
ρ =3
8πG
[γ2 − γ
tn+
k
c2e2γt
]+
1
2f c2, (7.136)
p =5γ
16πGtn− 3γ2
8πG− k
8πGc2e2γt− 1
2f c2. (7.137)
From equations (7.136) and (7.137), we obtain
ρ+ p = − γt−n
16πG+
1
4πG
k
c2e2γt(7.138)
and
ρ− p =3γ2
4πG− 11γt−n
16πG+
1
2πG
k
c2e2γt+ f c2. (7.139)
Again, from equations (7.136) and (7.137), we obtain
111
ρ+ 3p =9γt−n
16πG− 3γ2
4πG− f c2. (7.140)
From equations (7.136)-(7.140), it is observed that the NEC is satisfied if
c2 ≤ 4ke2γt[γt−n]
= A4, WEC is satisfied for c2 ≤ min
ke2γt[γt−n−γ2]
, 4ke2γt[γt−n]
= A5,
DEC is satisfied if c2 ≤ min
ke2γt[γt−n−γ2]
, 4ke2γt[γt−n]
, 8ke2γt[11γt−n−12γ2]
= A6 and
SEC is satisfied if 0 < c2 ≤ 9γ16πGftn
− 3γ2
4πGf.
It is observed that, for any value of t, NEC, WEC and DEC are satisfied in
this case if c2 ≤ min A4, A5, A6 whereas SEC is satisfied in this model for
0 < c2 ≤ 9γ16πGftn
− 3γ2
4πGf. However, we also observed that for large cosmic time ‘t’,
NEC, WEC and DEC are satisfied whereas SEC is violated.
Case 2: Power-law
Let a = αtδ, where δ > 1. Here α and δ are constants.
Now, Hubble parameter
H =a
a=αδtδ−1
αtδ=δ
t. (7.141)
From equations (7.124) and (7.141), we have
ρ =3
8πG
[δ2
t2+δ
tZt +
k
α2t2δ
]+
1
2f c2. (7.142)
From equations (7.124) and (7.141), we get
p =−3δ2 + 2δ
8πGt2− 5δZt
16πGt− 1
8πG
k
α2t2δ− 1
2f c2. (7.143)
From equations (7.142) and (7.143), we obtain
ρ+ p =1
4πG
δ
t2+
1
16πG
δ
tZt +
1
4πG
k
α2t2δ(7.144)
and
ρ− p =1
4πG
δ(3δ − 1)
t2+
11
16πG
δ
tZt +
1
2πG
k
α2t2δ+ f c2. (7.145)
Again, from equations (7.142) and (7.143), we obtain ρ+ 3p, where
112
ρ+ 3p =3
4πG
δ(1− δ)
t2− 9
16πG
δ
tZt − f c2. (7.146)
Subcase 2a: When Zt = −e−t
Substituting the value of Zt in equations (7.142) and (7.143), we can determine
ρ and p respectively as
ρ =3
8πG
[δ2
t2− δ
tet+
k
c2t2δ
]+
1
2f c2, (7.147)
p =2δ − 3δ2
8πGt2+
5δ
16πGtet− 1
8πG
k
c2t2δ− 1
2f c2. (7.148)
From equations (7.147) and (7.148), we obtain
ρ+ p =1
4πG
δ
t2− 1
16πG
δ
tet+
1
4πG
k
c2t2δ(7.149)
and
ρ− p =δ(3δ − 1)
4πGt2− 11
16πG
δ
tet+
1
2πG
k
c2t2δ+ f c2. (7.150)
Again, from equations (7.147) and (7.148), we obtain
ρ+ 3p =3δ(1− δ)
4πGt2+
9
16πG
δ
tet− f c2. (7.151)
From equations (7.147)-(7.151), it is noticed that the NEC is satisfied if
c2 ≤ 4kδt(2δ−1)e−t−4δ2t(2δ−2) = B1, WEC is satisfied for
c2 ≤ min
kδt2δ−1e−t−δ2t2δ−2 ,
4kδt(2δ−1)e−t−4δ2t(2δ−2)
= B2, DEC is satisfied if c2 ≤
min
kδt2δ−1e−t−δ2t2δ−2 ,
4kδt(2δ−1)e−t−4δ2t(2δ−2) ,
8k11δt2δ−1e−t−4δ(3δ−1)t2δ−2
= B3 and SEC is
satisfied if 0 < c2 ≤ 3δ(1−δ)4πGft2
+ 9δ16πGftet
.
It is observed that, for any value of t, NEC, DEC and SEC are satisfied in
this case if c2 ≤ min B1, B2, B3 whereas SEC is satisfied in this model for
0 < c2 ≤ 3δ(1−δ)4πGft2
+ 9δ16πGftet
. However, we also observed that for large cosmic time
‘t’, NEC, WEC and DEC are satisfied but SEC is violated.
113
Subase 2b: When Zt = −t−n
Substituting the value of Zt in equations (7.142) and (7.143), we can determine
ρ and p respectively as
ρ =3
8πG
[δ2
t2− δ
tn+1+
k
c2t2δ
]+
1
2f c2, (7.152)
p =2δ − 3δ2
8πGt2+
5δ
16πGtn+1− 1
8πG
k
c2t2δ− 1
2f c2. (7.153)
From equations (7.152) and (7.153), we obtain
ρ+ p =1
4πG
δ
t2− 1
16πG
δ
tn+1+
1
4πG
k
c2t2δ(7.154)
and
ρ− p =δ(3δ − 1)
4πGt2− 11
16πG
δ
tn+1+
1
2πG
k
c2t2δ+ f c2. (7.155)
Again, from equations (7.152) and (7.153), we obtain
ρ+ 3p =3δ(1− δ)
4πGt2+
9
16πG
δ
tn+1− f c2. (7.156)
From equations (7.116)-(7.120), it is observed that the NEC is satisfied if c2 ≤4k
δt2δ−n−1−δt2δ−2 = B4, WEC is satisfied for c2 ≤ min
kδt2δ−n−1−δ2t2δ−2 ,
4kδt2δ−n−1−δt2δ−2
=
B5, DEC is satisfied if c2 ≤ min
kδt2δ−n−1−δ2t2δ−2 ,
4kδt2δ−n−1−δt2δ−2 ,
8k11δt2δ−n−1−4δ(3δ−1)t2δ−2
=
B6 and SEC is satisfied if 0 < c2 ≤ 3δ(1−δ)4πGft2
+ 9δ16πGftn+1 .
It is observed that, for any value t, NEC, WEC and DEC are satisfied in this
case if c2 ≤ min B4, B5, B6 whereas SEC is satisfied in this model for 0 < c2 ≤3δ(1−δ)4πGft2
+ 9δ16πGftn+1 . However, we also observed that for large cosmic time ‘t’, NEC,
WEC and DEC are satisfied but SEC is violated.
7.6 Conclusions
In this chapter we have discussed the Finsler-Randers cosmological models in
modified theories of gravity. The present chapter dealt with the de Sitter solu-
114
tion, power law solution and general expansion solution. We have discussed the
behaviour of FR cosmological model in modified theory of gravity (Hoyle-Narlikar
creation field theory, Lyra geometry, General class of scalar-tensor theories and
Einstein theory) by considering Zt = −e−t and −t−n. We have also discussed
null energy condition (NEC), Weak energy condition (WEC), dominant energy
condition (DEC) and Strong energy condition (SEC) and find under what condi-
tions our FR cosmological model is physically stable in different modify theories
of gravitation. It is seen that all energy conditions are satisfied for some suitable
value of constant but for large cosmic time t (i.e. at present epoch), NEC, WEC
and DEC are satisfied but SEC is violated in all modify gravity theories, which is
responsible for current accelerated expansion of Universe. At t→ ∞ we obtained
Zt → 0, Finsler-Randers cosmological model tend to Friedman Robertson-Walker
model. The model represents an expanding Universe, which approaches isotropy
for large values of t. Results of this chapter are in favour of the observational
features of the Universe.
115
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List of Published/ Accepted/ Communicated Papers
List of Published Papers
1. Bankteshwar Tiwari and Manoj Kumar, On Randers change of a Finsler
space with m-th root metric, International Journal of Geometric Methods in
Modern Physics (World Scientific), Vol. 11 (2014) DOI: 10.1142/S021988781450087X.
2. Bankteshwar Tiwari and Manoj Kumar, Transformation of a Finsler Space
by Normalised Semi-Parallel Vector Fields, Journal of Tensor Society of
Japan, 74, 156-163, 2013.
List of Accepted Paper
1. Bankteshwar Tiwari and Manoj Kumar, On Finsler space with a special
(α, β)-metric, Journal of Indian Mathematical Society, (Accepted in 2014).
List of Communicated Papers
1. Bankteshwar Tiwari and Manoj Kumar, On Conformal Transformation
of m-th root Finsler metric, (Communicated).
2. Sapna Devi, Bankteshwar Tiwari and Manoj Kumar, Predator-prey model
with prey refuges: Jacobi stability versus Linear stability, (Communicated).
3. Bankteshwar Tiwari, Manoj Kumar and Ghanashyam Kumar Prajapati,
On the Projective change between two special Finsler spaces of (α, β)-metrics,
(Communicated).
4. R. Chaubey, Bankteshwar Tiwari, Anjani Kr. Shukla and Manoj Kumar,
Finsler-Randers Cosmological models in Modified Gravity Theories, (Com-
municated).
Papers presented in Conferences
1. International Conference on Differential Geometry and Relativity, (in collab-
oration with the Tensor Society) participated and presented a paper entitled
“On Finsler space with a special (α, β)-metric”, held at the Department of
Mathematics, Aligarh Muslim University, Aligarh, during November 20-22,
2012.
2. The 12th International Conference of Tensor Society (Japan) on Differential
Geometry and Its Applications, participated and presented a paper enti-
tled “Transformation of a Finsler Space by Normalised Semi-Parallel Vector
Fields”, held at the Department of Pure Mathematics, University of Cal-
cutta, during December 17-21, 2012.
3. International Conference on Differential Geometry and Relativity (ICDGR-
2013), presented a paper entitled “On Randers change of a Finsler space with
m-th root metric”, held at the Department of Mathematics and Statistics,
DDU Gorakhpur University, Gorakhpur, during November 09-11, 2013.
PERSONAL PROFILE
Manoj Kumar,
S/O Sri Harish Chandra Verma
Basdevpur, Bikapur
Distt.-Faizabad-224204
Uttar Pradesh, India.
Mobile: 91-9451788931
E-mail: mvermamath@gmail.com
Objective: To further continue my research and to get a teaching position in
some recognized institution offering opportunity for career advancement and pro-
fessional growth, which will help me to upgrade my knowledge and research work.
Educational Profile:
• M.Sc. (Mathematics) from Dr.R. M. L. Awadh University, Faizabad in 2008.
(Percentage: 62.86)• B.Sc. (Physics, Chemistry, Mathematics) from K. N. I. P. S. S., Sultanpur
in 2006. (Percentage: 57.83%)• Intermediate from G. V. I. C., Faizabad in 2003. (Percentage: 56.00%)• High school from G. V. I. C., Faizabad in 2001. (Percentage: 56.16%)
Software Skills: Ms-office, Maple, Matlab, Mathematica and LaTeX.
Area of Interest: Finsler Geometry, Modelling and Cosmology.
Declaration:
I hereby declare that the information furnished above is true to the best of my
knowledge.
Manoj Kumar
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