Killing Vector Fields in Generalized Conformal β-Change of Finsler Spaces

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.


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  (, ) →  (, ) =  ( ()  (, ) ,  (, )) , (1) 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 ( =   , ).
The angular metric tensor ℎ  of the space   is given by [10] where ℎ  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 (ℎ)ℎV-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] where The symbol "|" denotes ℎ-covariant derivative with respect to Cartan connection Γ and lower index "0" (except in  0 ) 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] where The coefficients    of Cartan connection Γ in   and the coefficients    of the corresponding Cartan connection Γ in   are related as follows [10]: where The tensor    has the properties where    =      . (17)

Killing Vector Fields in
Correspondence of   and

𝑛
Let us consider an infinitesimal transformation where  is an infinitesimal constant and V  () is a contravariant vector field.The vector field V  () 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 m V being the operator of Lie differentiation.Equivalently, the vector field V  () is Killing in   if where 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 and only if where    is the associate Cartan tensor of   .
Proof.Assume that V  () is Killing in   .Then (20) is satisfied.By definition, the ℎ-covariant derivatives of V  with respect to Γ and Γ are, respectively, given as where   = /  and "‖" denote the ℎ-covariant differentiation with respect to Γ.Equation ( 22)(a), by virtue of ( 11), (15), and (22)(b), takes the form Now, from (23), we have Using ( 9) in (24) and applying (20), we get Proof is complete with the observation that V  () is Killing in As another important consequence of Theorem 1, we have the following.Corollary 3. If a vector field V  () is Killing in   and   , then the vector V  (, ) is orthogonal to the vector   (, ).

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 V  (), generating a motion in   , to generate a motion in   as well.It is clear that vector field V  (), 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   .