Abstract

We consider a Finsler space equipped with a Generalized Conformal β-change of metric and study the Killing vector fields that correspond between the original Finsler space and the Finsler space equipped with Generalized Conformal β-change of metric. We obtain necessary and sufficient condition for a vector field Killing in the original Finsler space to be Killing in the Finsler space equipped with Generalized Conformal β-change of metric.

1. Introduction

In 1976, Hashiguchi [1] studied the conformal change of Finsler metrics; namely, . In particular, he also dealt with the special conformal transformation named -conformal transformation. This change has been studied by Izumi [2] and Kropina [3]. In 2008, Abed [4, 5] introduced the transformation , thus generalizing the conformal, Randers, and generalized Randers changes. Moreover, he established the relationships between some important tensors associated with and the corresponding tensors associated with . He also studied some invariant and -invariant properties and obtained a relationship between the Cartan connection associated with and the transformed Cartan connection associated with .

In this paper, we deal with a general change of Finsler metrics defined by where is a positively homogeneous function of degree one in and . This change will be referred to as a generalized -conformal change. It is clear that this change is a generalization of the abovementioned changes and deals simultaneously with -change and conformal change. It combines also the special case of Shibata and that of Abed .

In 1984, Shibata [6] studied -change of Finsler metrics and discussed certain invariant tensors under such a change. Killing equations play important role in the study of a Finsler space which undergoes a change in the metric. In fact, they give an equivalent characterization for the transformations to preserve distances. In 1979, Singh et al. [7] studied a Randers space , , which undergoes a change . They discussed Killing correspondence of spaces and .

In the present paper, we consider a general Finsler space which undergoes conformal and -change; that is, , where is a 1-form. We study Killing correspondence of Finsler spaces and . For the notations and terminology, we refer the reader to the books [8, 9] and the papers [6] by Shibata and [10] by Youssef et al.

2. Preliminaries

Let , , be an -dimensional Finsler manifold with fundamental function . Consider the following change of Finsler structures which will be referred to as a generalized -conformal change:where is a positively homogeneous function of degree one in and -form , where .

We define where .

The angular metric tensor of the space is given by [10]where is the angular metric tensor of . The fundamental metric tensor and its inverse of are expressed as [10]where and , respectively, are the metric tensor and inverse metric tensor of . The Cartan tensor and the associate Cartan tensor of are given by the following expressions:The -torsion tensor is expressed in terms of as [10]where and are, respectively, the Cartan tensor and the associate Cartan tensor of . The spray coefficients of in terms of the spray coefficients of are expressed as [10]whereThe symbol “” denotes -covariant derivative with respect to Cartan connection and lower index “” (except in ) denotes the contraction by .

The relation between the coefficients of Cartan nonlinear connection in and the coefficients of the corresponding Cartan nonlinear connection in is given by [10]whereThe coefficients of Cartan connection in and the coefficients of the corresponding Cartan connection in are related as follows [10]:whereThe tensor has the properties

3. Killing Vector Fields in Correspondence of and

Let us consider an infinitesimal transformationwhere is an infinitesimal constant and is a contravariant vector field.

The vector field is said to be a Killing vector field in if the metric tensor of the Finsler space with respect to the infinitesimal transformation (18) is Lie invariant; that is,with being the operator of Lie differentiation. Equivalently, the vector field is Killing in ifwhere .

Now, we prove the following result which gives a necessary and sufficient condition for a Killing vector field in to be Killing in .

Theorem 1. A Killing vector field in is Killing in if and only ifwhere is the associate Cartan tensor of .

Proof. Assume that is Killing in . Then (20) is satisfied. By definition, the -covariant derivatives of with respect to and are, respectively, given aswhere and “” denote the -covariant differentiation with respect to . Equation (22)(a), by virtue of (11), (15), and (22)(b), takes the formNow, from (23), we haveUsing (9) in (24) and applying (20), we getProof is complete with the observation that is Killing in if and only if , that is, if and only if (21) holds.

If a vector field is Killing in and , then, from Theorem 1, (21) holds, which on transvection by yieldsEquation (21), in view of (26), enables us to state the following.

Corollary 2. If a vector field is Killing in and , then

As another important consequence of Theorem 1, we have the following.

Corollary 3. If a vector field is Killing in and , then the vector is orthogonal to the vector .

Proof. As is Killing in and , (21) holds, which on transvection by gives (26). Again Transvection (26) by , it follows that . This proves the result.

4. Conclusion

The main purpose of the present paper is to examine the classical approach to the problem of existence of Killing vector fields and study how they vary from point to point and how they are related to Killing vector fields defined on the whole manifold. In this respect, our purpose is similar to that of Shukla and Gupta on the study of projective motion. Actually, there is a more substantial relation of our work to theirs where we proved Theorem 1 as the main result and as its consequences we obtained Corollaries 2 and 3. Since the Killing equation (19) is a necessary and sufficient condition for the transformation (18) to be a motion in , condition (21) obtained in Theorem 1 may be taken as the necessary and sufficient condition for the vector field , generating a motion in , to generate a motion in as well. It is clear that vector field , generating an affine motion (resp., projective motion) in , generates an affine motion (resp., projective motion) in if condition (21) holds. Our study has applications to link various transformations in with the corresponding transformations in .

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.