Abstract

A special form of () -torsion tensor was introduced which may be considered generalization of -Finsler space and -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

1. Introduction

Let be an -dimensional differentiable manifold and be its tangent bundle. The manifold is covered by neighborhoods (), in each of which we have a local coordinate system . A tangent vector at a point of is written as , and we have a local coordinate system of over .

In paper [1] let be an -dimensional Finsler space with metric function . There are five kinds of function . There are five kinds of torsion tensors in the theory of Finsler space based on Carton’s connection, out of whichas () -torsion tensor and () -Torsion tensor are of great important tensors for the present study, where is h-curvature tensor. In Finsler geometry based on Cartan’s connection, there are three kinds of covariant differentiations denoted as and v-covariant differentiation denoted as .

An -dimensional Finsler space is said to be a semi--reducible Finsler space, whose Cartan’s tensor is written as where and scalars satisfy . Moreover if scalars and are constants, is said to be -reducible Finsler space with constants coefficients.

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

An -dimensional Finsler space is said to be a Special semi--reducible Finsler space (in short we call SSR-Finsler space) [1, 2] whose ()-torsion tensor is written as Various interesting forms of these tensors have been studied by many ([3–7], …), two of them are -reducible Finsler space and a Special semi--reducible Finsler space ([1, 2]) in which the torsion tensor , respectively, is in the formswhere is angular metric tenser and , where is reciprocal of the metric tensor

Izumi ([4, 5]) introduced -Finsler space in which is of the form where is the scalar homogeneous function on of zero degree in . In -reducible Finsler space the tensor is the form [8]where . A Finsler space with is called a Landsbergs space [9]. If then is called Berwald’s affinely connected space ([10, 11]).

Rund [11] introduced a special form of torsion tensor as follows:where is a scalar homogeneous function on of degree 1 and is a homogeneous function of degree 0 with respect to . He then studied some properties of satisfying (8). The present author introduced a more general form of (8) and studies some properties of satisfying it [12].

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

Lemma 1 (see [6]). If the curvature tensor of a -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 is locally Minkowskian if h-curvature tensor and .

Definition 3 (see [1]). A Finsler connection is defined as tried as -connection and -connection which are components of a tensor field of (1, 2)-type. The tensor of component is called deflection tensor of . Therefore, and are desirable conditions for a Finsler connection.
Let be an -dimensional modified by (we mean the tangent space at ) and by (we mean the slit tangent bundle of ).

A Finsler metric on is a function which has the following properties:(i) is .(ii) is positively homogeneous function of degree 1 on .(iii)For each , the metric tensor and the angular metric tensor are, respectively, given by The angular metric tensor can also be written in terms of the normalized element of support(see [14]). For, Cartan’s tensor vector is defined asAccording to Deicke’s theorem is the necessary and sufficient condition for to be Riemannian. Let be a Finsler space for . We define Matsumoto torsions of -reducible and Special semi--reducible Finsler space, respectively, as follows:A Finsler space is said to be -reducible if and is Special semi--reducible if  Next, we define a tensor where “∣” means h-covariant differentiation with respect to Cartan’s connection.

A Finsler space is called a Landsberg space if , or equivalently .

DefineA Finsler space is said to be weakly Landsberg space if [15].

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

In paper [1] definewhere Let ; hence where , , and are some scalar function homogeneous of degree 1 and ’s are homogeneous of degree zero. It is obvious that is a -reducible Finsler space if .

The purpose of the present paper is to study satisfying (18).

If is a Landsberg space then ; hence from (18)where

Corollary 4. A Landsbergs space satisfying (18) is a Special semi--reducible Finsler space.
Since for to be Landsberg space , therefore from Lemma 1 and Corollary 4.

Corollary 5. A Landsbergs space satisfying (18) is Berwald’s affinely connected space, if
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, , then it is locally Minkowskian.

Special Forms of . Let be a Finsler space satisfying (18). A Finsler space with of given form reduces to -Finsler space when and , while it reduces to -reducible Finsler space when and and .

By definition from (18) we can write Contracting by we get By replacing (23) into (22)orwhere Hence we have the following.

Theorem 7. The Matsumoto torsion of -reducible Finsler space and Matsumoto torsion of Special semi--reducible Finsler space are related by

Corollary 8. A Finsler space satisfying (18) is a weakly Landsberg space ifThe notation of stretch curvature denoted by was introduced by Berwald as generalization of Landsberg curvature [10] in which A Finsler space is said to be stretch space if .

Again taking h-covariant derivative of (22) and then contracting by , we get where we put , , and

Suppose that is stretch space; then By contacting (30) with , we obtain From (32) and (30) we haveContacting by (33) by whence Substituting (35) into (33), we getFrom (36), it follows that is a semi--reducible Finsler space if it is a weakly Landsberg space.

Therefore we have the following.

Theorem 9. Let a Finsler space satisfying (18) be a stretch space; then it is a Special semi--reducible Finsler space, if it is a weakly Landsberg space.

2-Connection. A connection connects with tengent spaces of two points of manifold. The quantities are connection coefficients if (see [16]).

Connection. is uniquely expressible as the sum of the symmetric connections and the torsion tensor [12] whereis symmetric connection A connection is called symmetric connection if .

Torsion tensor for symmetric connection science isthat is, is a skew-symmetric tensor.

Five kinds of torsion tensors [17] are as follows:It is noted that -connection also plays a role of torsion tensor and(see [9]).From (40) and (43) we haveFrom (38) and (46), we haveFrom (39b) and (43) we haveFor Carton’s connection () torsion .

Hence from (47) we haveUsing (45)Also for () -torsion from (47)Then tensor of component is called the deflection tensor .

ThereforePut ; we haveUsing (52) we have

Theorem 10. For Cartan’s connection (h) h tensor and deflection FT is symmetric as (52).

Competing Interests

The authors declare that they have no competing interests.