On a Special Form of ( h ) h [nu] -Torsion Tensor P i j k in Finsler Space

Abstract

A special form of ( h ) h ν -torsion tensor was introduced which may be considered generalization of [superscript] P * [/superscript] -Finsler space and P -reducible Finsler space and then some properties of this space were studied. We also introduce connection and give some case and condition of torsion tensor [superscript] T j k i [/superscript] .

Brijesh Kumar Tripathi 1 and K. B. Pandey 2

Academic Editor:Willi Freeden

1, Department of Mathematics, L.E. College, Morbi, Gujarat 363642, India

2, Department of Mathematics, K.N.I.T., Sultanpur 228118, India

Received 11 May 2016; Revised 18 August 2016; Accepted 29 August 2016

This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

1. Introduction

Let ^{M n} be an n -dimensional differentiable manifold and T ^{M n} be its tangent bundle. The manifold ^{M n} is covered by neighborhoods ( U ), in each U of which we have a local coordinate system ^{( x i} ) . A tangent vector at a point x = ( ^{x i} ) of U is written as ^{y i} ( ∂ / ∂ ^{x i} ) x , and we have a local coordinate system ( ^{x i} , ^{y i} ) of T ^{M n} over U .

In paper [1] let ^{F n} ( n ≥ 1 ) be an n -dimensional Finsler space with metric function L ( x , y ) . There are five kinds of function L ( x , y ) . There are five kinds of torsion tensors in the theory of Finsler space based on Carton's connection, out of which [figure omitted; refer to PDF] as ( ν ) h ν -torsion tensor and ( h ) h -Torsion tensor are of great important tensors for the present study, where _{P h i j k} is h ν -curvature tensor. In Finsler geometry based on Cartan's connection, there are three kinds of covariant differentiations denoted as _{| i} and v-covariant differentiation denoted as | i .

An n ( n ≥ 3 ) -dimensional Finsler space ^{F n} is said to be a semi- C -reducible Finsler space, whose Cartan's tensor _{C i j k} is written as [figure omitted; refer to PDF] where ^{C 2} = ^{g i j} _{C i C j} and scalars satisfy p + q = 1 . Moreover if scalars p and q are constants, ^{F n} is said to be C -reducible Finsler space with constants coefficients.

A Special semi- C -reducible Finsler space has been introduced by Ikeda [2] as follows.

An n ( n ≥ 3 ) -dimensional Finsler space ^{F n} is said to be a Special semi- C -reducible Finsler space (in short we call SSR-Finsler space) [1, 2] whose h ( h ν )-torsion tensor _{C i j k} is written as [figure omitted; refer to PDF] Various interesting forms of these tensors have been studied by many ([3-7], ...), two of them are C -reducible Finsler space and a Special semi- C -reducible Finsler space ([1, 2]) in which the torsion tensor _{C i j k} , respectively, is in the forms [figure omitted; refer to PDF] where _{h i j} is angular metric tenser and _{C i} = _{C i j k} ^{g j k} , where ^{g j k} is reciprocal of the metric tensor _{g j k} .

Izumi ([4, 5]) introduced ^{P [low *]} -Finsler space in which _{P i j k} is of the form [figure omitted; refer to PDF] where λ is the scalar homogeneous function on T M of zero degree in ^{y i} . In P -reducible Finsler space the tensor _{P i j k} is the form [8] [figure omitted; refer to PDF] where _{J i} = _{C i |" 0} = _{C i |" j} ^{y i} . A Finsler space with _{P i j k} = 0 is called a Landsbergs space [9]. If _{C i j k |" h} = 0 then ^{F n} is called Berwald's affinely connected space ([10, 11]).

Rund [11] introduced a special form of torsion tensor _{P i j k} as follows: [figure omitted; refer to PDF] where λ = λ ( x , y ) is a scalar homogeneous function on T M of degree 1 and _{a i} = _{a i} ( x ) is a homogeneous function of degree 0 with respect to ^{y i} . He then studied some properties of ^{F n} satisfying (8). The present author introduced a more general form of (8) and studies some properties of ^{F n} satisfying it [12].

We quote the following lemmas, which will be used in the present paper.

Lemma 1 (see [6]).

If the curvature tensor _{P i j k} of a C -reducible Finsler space vanishes then the space vanishes and then the space is Berwald's affinely connected space.

Lemma 2 (see [13]).

A Finsler space _{F n} is locally Minkowskian if h-curvature tensor _{R h i j k} = 0 and _{C i j k |" h} = 0 .

Definition 3 (see [1]).

A Finsler connection F Γ is defined as tried ( ^{F j k i} ( x , y ) , ^{V j k i} ( x , y ) ) as h -connection and v -connection which are components of a tensor field of (1, 2)-type. The tensor D of component ^{D j i} is called deflection tensor of F Γ . Therefore, ^{D j i = N j i} and ^{y r V r j i} = 0 are desirable conditions for a Finsler connection.

Let M be an n -dimensional ^{C ∞} modified by _{T x} M (we mean the tangent space at x ∈ M ) and by T M \ 0 (we mean the slit tangent bundle of M ).

A Finsler metric on M is a function L : T M [arrow right] [ 0 , ∞ ) which has the following properties:

(i) L is ^{C ∞} o n T M \ 0 .

(ii) L is positively homogeneous function of degree 1 on T M .

(iii): For each y ∈ _{T x} M , the metric tensor _{g i j} and the angular metric tensor _{h i j} are, respectively, given by [figure omitted; refer to PDF]

The angular metric tensor _{h i j} can also be written in terms of the normalized element of support [figure omitted; refer to PDF] (see [14]). For y ∈ _{T x} M \ 0 , Cartan's tensor vector is defined as [figure omitted; refer to PDF] According to Deicke's theorem _{C i} = 0 is the necessary and sufficient condition for ^{F n} to be Riemannian. Let ^{F n} = ( ^{M n} , L ) be a Finsler space for y ∈ _{T x} M \ 0 . We define Matsumoto torsions of C -reducible and Special semi- C -reducible Finsler space, respectively, as follows: [figure omitted; refer to PDF] A Finsler space ^{F n} is said to be C -reducible if _{M i j k} = 0 and is Special semi- C -reducible

: if [figure omitted; refer to PDF]

: Next, we define a tensor [figure omitted; refer to PDF]

: where "|"" means h-covariant differentiation with respect to Cartan's connection.

A Finsler space ^{F n} is called a Landsberg space if _{P i j k} = 0 , or equivalently _{L i j k} = _{C i j k |" h} = 0 .

Define [figure omitted; refer to PDF] A Finsler space is said to be weakly Landsberg space if _{J i} = 0 [15].

It is obvious that every C -reducible Finsler space is P -reducible, but the converse is not true.

In paper [1] define [figure omitted; refer to PDF] where [figure omitted; refer to PDF] Let B = - ^{b [low *]} ; hence [figure omitted; refer to PDF] where λ , A , and B are some scalar function homogeneous of degree 1 and _{a i} 's are homogeneous of degree zero. It is obvious that ^{F n} is a P -reducible Finsler space if _{M i j k} = 0 .

The purpose of the present paper is to study ^{F n} satisfying (18).

If ^{F n} is a Landsberg space then _{P i j k} = 0 ; hence from (18) [figure omitted; refer to PDF] where [figure omitted; refer to PDF]

Corollary 4.

A Landsbergs space satisfying (18) is a Special semi- C -reducible Finsler space.

Since for ^{F n} to be Landsberg space _{P i j k h} = 0 , therefore from Lemma 1 and Corollary 4.

Corollary 5.

A Landsbergs space satisfying (18) is Berwald's affinely connected space, if B = 0 .

In view of Lemma 2 and Corollary 5 one has the following.

Corollary 6.

If Landsbergs space satisfying (18) has vanishing h-curvature tensor, that is, _{R i j k h} = 0 , then it is locally Minkowskian.

Special Forms of _{P i j k} . Let ^{F n} be a Finsler space satisfying (18). A Finsler space with _{P i j k} of given form reduces to ^{P [low *]} -Finsler space when A = 0 and B = 0 , while it reduces to P -reducible Finsler space when λ = 0 and B = 0 and A _{a i} = ( 1 / ( n + 1 ) ) _{J i} .

By definition from (18) we can write [figure omitted; refer to PDF] Contracting by ^{g i j} we get [figure omitted; refer to PDF] By replacing (23) into (22) [figure omitted; refer to PDF] or [figure omitted; refer to PDF] where [figure omitted; refer to PDF] Hence we have the following.

Theorem 7.

The Matsumoto torsion of P -reducible Finsler space _{M i j k} and Matsumoto torsion of Special semi- C -reducible Finsler space _{M - i j k} are related by [figure omitted; refer to PDF]

Corollary 8.

A Finsler space ^{F n} satisfying (18) is a weakly Landsberg space if [figure omitted; refer to PDF] The notation of stretch curvature denoted by _{Σ h i j k} was introduced by Berwald as generalization of Landsberg curvature [10] in which [figure omitted; refer to PDF] A Finsler space is said to be stretch space if _{Σ h i j k} = 0 .

Again taking h-covariant derivative of (22) and then contracting by ^{y h} , we get [figure omitted; refer to PDF] where we put λ - = _{λ | h} , A - = _{A | h} , and B - = _{B | h} .

Suppose that ^{F n} is stretch space; then [figure omitted; refer to PDF] By contacting (30) with ^{y k} , we obtain [figure omitted; refer to PDF] From (32) and (30) we have [figure omitted; refer to PDF] Contacting by (33) by ^{g j h} [figure omitted; refer to PDF] whence [figure omitted; refer to PDF] Substituting (35) into (33), we get [figure omitted; refer to PDF] From (36), it follows that ^{F n} is a semi- C -reducible Finsler space if it is a weakly Landsberg space.

Therefore we have the following.

Theorem 9.

Let a Finsler space ^{F n} satisfying (18) be a stretch space; then it is a Special semi- C -reducible Finsler space, if it is a weakly Landsberg space.

2-Connection . A connection connects with tengent spaces of two points of manifold. The ^{n 3} quantities ^{L j k i} are connection coefficients if [figure omitted; refer to PDF] (see [16]).

Connection . ^{L j k i} is uniquely expressible as the sum of the symmetric connections and the torsion tensor [12] [figure omitted; refer to PDF] where [figure omitted; refer to PDF] is symmetric connection [figure omitted; refer to PDF] A connection ^{L j k i} is called symmetric connection if ^{L j k i} = ^{L k j i} .

Torsion tensor for symmetric connection science is [figure omitted; refer to PDF] that is, ^{T j k i} is a skew-symmetric tensor.

Five kinds of torsion tensors [17] are as follows: [figure omitted; refer to PDF] It is noted that ν -connection ( ^{V j k i} ) also plays a role of torsion tensor and [figure omitted; refer to PDF] (see [9]). [figure omitted; refer to PDF] From (40) and (43) we have [figure omitted; refer to PDF] From (38) and (46), we have [figure omitted; refer to PDF] From (39b) and (43) we have [figure omitted; refer to PDF] For Carton's connection ( h ) torsion ^{T j k i} = 0 .

Hence from (47) we have [figure omitted; refer to PDF] Using (45) [figure omitted; refer to PDF] Also for ( h ) h -torsion ^{S j k i} = 0 from (47) [figure omitted; refer to PDF] Then tensor D of component ^{D j i} is called the deflection tensor F .

Therefore [figure omitted; refer to PDF] Put r = k ; we have [figure omitted; refer to PDF] Using (52) we have [figure omitted; refer to PDF]

Theorem 10.

For Cartan's connection (h) h tensor ^{S j k i} and deflection FT ^{V j k i} is symmetric as (52).

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