# Killing Vector Fields in Generalized Conformalβ-Change of Finsler Spaces

Authors: Mallikarjun Yallappa Kumbar; Narasimhamurthy Senajji Kampalappa; Thippeswamy Komalobiah Rajanna; Kavyashree Ambale Rajegowda
Journal: Journal of Mathematics
 (2015)

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## Abstract

We consider a Finsler space equipped with a Generalized Conformalβ-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.


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## Body

## 1. Introduction

In 1976, Hashiguchi [1] studied the conformal change of Finsler metrics; namely, L¯=eσ(x)L. In particular, he also dealt with the special conformal transformation named C-conformal transformation. This change has been studied by Izumi [2] and Kropina [3]. In 2008, Abed [4, 5] introduced the transformation L¯=eσ(x)L+β, thus generalizing the conformal, Randers, and generalized Randers changes. Moreover, he established the relationships between some important tensors associated with (M,L) and the corresponding tensors associated with (M,L¯). He also studied some invariant and σ-invariant properties and obtained a relationship between the Cartan connection associated with (M,L) and the transformed Cartan connection associated with (M,L¯).In this paper, we deal with a general change of Finsler metrics defined by(1)Lx,y⟶L¯x,y=feσxLx,y,βx,y,where f is a positively homogeneous function of degree one in L¯≔eσL and β. This change will be referred to as a generalized β-conformal change. It is clear that this change is a generalization of the abovementioned changes and deals simultaneously with β-change and conformal change. It combines also the special case of Shibata (L¯=f(L,β)) and that of Abed (L¯=eσL,β).In 1984, Shibata [6] studied β-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 Fn(M,L(x,y)=(gij(x)yiyj)1/2+bi(x)yi), n≥2, which undergoes a change L(x,y)↦L∗(x,y)=L2(x,y)+(αi(x)yi)2. They discussed Killing correspondence of spaces Fn(M,L) and F∗n(M,L∗).In the present paper, we consider a general Finsler spaceFn(M,L) which undergoes conformal and β-change; that is, L(x,y)→L¯(x,y)=f(eσ(x)L(x,y),β(x,y)), where β(x,y)=bi(x)yi is a 1-form. We study Killing correspondence of Finsler spaces Fn(M,L) and F¯n(M,L¯). 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

LetFn=(M,L), n≥2, be an n-dimensional C∞ Finsler manifold with fundamental function L=L(x,y). Consider the following change of Finsler structures which will be referred to as a generalized β-conformal change:(2)Lx,y⟶L¯x,y=feσxLx,y,βx,y,where f is a positively homogeneous function of degree one in eσL and 1-form β, where β=bi(x)dxi.We define(3)f1≔∂f∂L~,f2≔∂f∂β,f12≔∂2f∂L~∂β,…,where L~=eσL.The angular metric tensorh¯ij of the space F¯n is given by [10](4)h¯ij=eσphij+q0mimj,where(5)p=ff1L,q=ff2,q0=ff22,p0=f22+q0,q-1=ff12L,p-1=q-1+pf2f,q-2=feσf11-f1/LL2,p-2=q-2+eσp2f2,mi=bi-βyiL2≠0,σi=∂iσ.hij is the angular metric tensor of Fn. The fundamental metric tensor g¯ij and its inverse g¯ij of F¯n are expressed as [10](6)g¯ij=eσpgij+p0bibj+eσp-1biyj+bjyi+eσp-2yiyj,g¯ij=e-σpgij-s0bibj-s-1biyj+bjyi-s-2yiyj,where(7)s0=e-σf2q0ɛpL2,s-1=p-1f2ɛpL2,s-2=p-1eσm2pL2-b2f2ɛpβL2,ɛ=f2eσp+m2q0L2≠0,m2=gijmimj.gij and gij, respectively, are the metric tensor and inverse metric tensor of Fn. The Cartan tensor C¯ijk and the associate Cartan tensor C¯ijl of F¯n are given by the following expressions:(8)C¯ijk=eσpCijk+12eσp-1hijmk+hjkmi+hkimj+12p02mimjmk.The (h)hv-torsion tensor C¯ijl is expressed in terms of Cijl as [10](9)C¯ijl=Cijl+Mijl,where(10)Mijl=12pe-σml-pm2s0bl+s-1yleσp-1hij+p02mimj-eσs0bl+s-1ylpCisjbs+p-1mimj+p-12philmj+hjlmi;hji=gilhlj,p02=∂p0∂β.Cijk and Cijl are, respectively, the Cartan tensor and the associate Cartan tensor of Fn. The spray coefficients G¯i of F¯n in terms of the spray coefficients Gi of Fn are expressed as [10](11)G¯i=Gi+Di,where(12)Di=σ02p2p-βp-1-eσp2L2s-2-ps-12eσpβ+eσp-1L2m2yi-2eσp2βs0bi+qpe-σF0i-12L2σi+12eσpE00-2qFβ0+eσpL2σβs0bi+s-1yi;Ejk=12bj∣k+bk∣j,Fjk=12bj∣k-bk∣j,Fji=gikFkj.The symbol “∣” denotes h-covariant derivative with respect to Cartan connection CΓ and lower index “0” (except in s0) denotes the contraction by yi.The relation between the coefficientsN¯ji of Cartan nonlinear connection in F¯n and the coefficients Nji of the corresponding Cartan nonlinear connection in Fn is given by [10](13)N¯ji=Nji+Dji,where(14)Dji=e-σpAji-s0bi+s-1yiAtjbt-qb0∣j+eσpL2σjs-1bi+s-2yi;Aij=E00Bij+Fi0Qj+qFij+Ej0Qi-2eσpCsij+VsijDs+12σ02eσpgij+2eσp-1mjyi-2βBij+eσp-1biyj-bjyi-12σieσL2p-1mj+2eσpyj+12σj2eσpyi+eσL2p-1mi;Aji=gliAlj,2Bij=eσp-1hij+p02mimj,Qi=eσp-1yi+p0bi.The coefficients F¯jki of Cartan connection CΓ¯ in F¯n and the coefficients Fjki of the corresponding Cartan connection CΓ in Fn are related as follows [10]:(15)F¯jki=Fjki+Djki,where(16)Djki=e-σpgit-s0bi+s-1yibt-s-1bi+s-2yiytFtkQj+FtjQk+EjkQt+12Θj,k,t2eσpCjkmDtm+2VjkmDtm-Kjkσt-2Bjkb0∣t,Vijk=12eσp-1hijmk+hjkmi+hkimj+12p02mimjmk,Kij=A1gij+A2bibj+A3biyj+bjyi+A4yiyj,A1=eσ2p-βp-1,A2=-βp02,A3=eσp-1+β2L2p02,A4=eσp-2-β3L4p02,Θj,k,tAjkt=Ajkt-Aktj-Atjk.The tensor Djki has the properties(17)Dj0i=Bj0i=Dji;D00i=2Di,whereBjki=∂kDji.


## 3. Killing Vector Fields in Correspondence ofFn and F¯n

Let us consider an infinitesimal transformation(18)x′i=xi+ϵvix,where ϵ is an infinitesimal constant and vi(x) is a contravariant vector field.The vector fieldvi(x) is said to be a Killing vector field in Fn if the metric tensor of the Finsler space with respect to the infinitesimal transformation (18) is Lie invariant; that is,(19)£vgij=0,with £v being the operator of Lie differentiation. Equivalently, the vector field vi(x) is Killing in Fn if(20)vi∣j+vj∣i+2Cijlvl∣0=0,where vi=gilvl.Now, we prove the following result which gives a necessary and sufficient condition for a Killing vector field inFn to be Killing in F¯n.Theorem 1.
A Killing vector fieldvi(x) in Fn is Killing in F¯n if and only if(21)Mijlvl∣0+CrjtvtDir+CritvtDjr+vrDijr+C¯ijl2CrltvtDr+vrDlr=0,where C¯ijl is the associate Cartan tensor of F¯n.Proof.
Assume thatvi(x) is Killing in Fn. Then (20) is satisfied. By definition, the h-covariant derivatives of vi with respect to CΓ¯ and CΓ are, respectively, given as(22)avi∥j=∂jvi-∂˙rviG¯jr-vrF¯ijr,bvi∣j=∂jvi-∂˙rviGjr-vrFijr,where ∂j=∂/∂xj and “∥” denote the h-covariant differentiation with respect to CΓ¯. Equation (22)(a), by virtue of (11), (15), and (22)(b), takes the form(23)vi∥j=vi∣j-2CritvtDjr-vrDijr.Now, from (23), we have(24)vi∥j+vj∥i+2C-ijlvl∥0=vi∣j+vj∣i+2Cijlvl∣0-2CritvtDjr-2CrjtvtDir-2vrDijr-2C-ijl2CrltvtDr+vrDlr.Using (9) in (24) and applying (20), we get(25)vi∥j+vj∥i+2C-ijlvl∥0=2Mijlvl∣0-2CritvtDjr-2CrjtvtDir-2vrDijr-2C-ijl2CrltvtDr+vrDlr.Proof is complete with the observation that vi(x) is Killing in F¯n if and only if vi∥j+vj∥i+2C¯ijlvl∥0=0, that is, if and only if (21) holds.If a vector fieldvi(x) is Killing in Fn and F¯n, then, from Theorem 1, (21) holds, which on transvection by yi yields(26)2CrltvtDr+vrDlr=0.Equation (21), in view of (26), enables us to state the following.Corollary 2.
If a vector fieldvi(x) is Killing in Fn and F¯n, then(27)CritvtDjr+CrjtvtDir+vrDijr-Mijlvl∣0=0.As another important consequence of Theorem1, we have the following.Corollary 3.
If a vector fieldvi(x) is Killing in Fn and F¯n, then the vector vi(x,y) is orthogonal to the vector Di(x,y).Proof.
Asvi(x) is Killing in Fn and F¯n, (21) holds, which on transvection by yi gives (26). Again Transvection (26) by yj, it follows that vrDr=0. 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 Theorem1 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 Fn, condition (21) obtained in Theorem 1 may be taken as the necessary and sufficient condition for the vector field vi(x), generating a motion in Fn, to generate a motion in F¯n as well. It is clear that vector field vi(x), generating an affine motion (resp., projective motion) in Fn, generates an affine motion (resp., projective motion) in F¯n if condition (21) holds. Our study has applications to link various transformations in Fn with the corresponding transformations in F¯n.



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*Source: 456291-2015-10-26.xml*