Kyungpook Mathematical Journal 2018; 58(4): 781-788  
Finslerian Hypersurface and Generalized β–Conformal Change of Finsler Metric
Shiv Kumar Tiwari*, and Anamika Rai
Department of Mathematics, K. S. Saket Post Graduate College, Ayodhya, Faizabad224 123, India
e-mail : sktiwarisaket@yahoo.com and anamikarai2538@gmail.com
*Corresponding Author.
Received: March 11, 2015; Accepted: February 13, 2018; Published online: December 23, 2018.
© Kyungpook Mathematical Journal. All rights reserved.

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Abstract

In the present paper, we have studied the Finslerian hypersurfaces and generalized β–conformal change of Finsler metric. The relations between the Finslerian hypersurface and the other which is Finslerian hypersurface given by generalized β–conformal change have been obtained. We have also proved that generalized β–conformal change makes three types of hypersurfaces invariant under certain conditions.

Keywords: generalized β-conformal change, generalized β-change, β-change, conformal change, Finslerian hypersurfaces, hyperplane of fi rst, second and third kinds.
1. Introduction

Let (Mn, L) be an n–dimensional Finsler space on a differentiable manifold Mn equipped with the fundamental function L(x, y). In 1984, Shibata [12] introduced the transformation of Finsler metric:

L¯(x,y)=f(L,β),

where β = bi(x) yi, bi(x) are components of a covariant vector in (Mn, L) and f is positively homogeneous function of degree one in L and β. This change of metric is called a β–change. In 2013, Prasad, B. N. and Kumari, Bindu [10] have considered the β–change of Finsler metric. In the year 2014 [13], we studied generalized β–change defining as

L(x,y)L¯(x,y)=f(L,β1),β2),,βm)),

where f is any positively homogeneous function of degree one in L, β1), β2), …, βm), where β1), β2), …, βm) are linearly independent one-form.

The conformal theory of Finsler spaces has been initiated by M. S. Knebelman [7] in 1929 and has been investigated in detail by many authors [1, 2, 3, 6] etc. The conformal change is defined as

L(x,y)eσ(x)L(x,y),

where σ(x) is a function of position only and known as conformal factor.

We also studied the generalized β–conformal change of Finsler metric by taking

L¯=f(eσ(x)L(x,y),β1),β2),,βm)),

where f is any positively homogeneous function of degree one in eσL, β1), β2), …, βm).

On the other hand, in 1985, M. Matsumoto investigated the theory of Finslerian hypersurface [8]. He has defined three types of hypersurfaces that were called a hyperplane of the first, second and third kinds.

In the year 2009, B. N. Prasad and Gauri Shanker [11] studied the Finslerian hypersurfaces and β–change of Finsler metric and obtained different results in his paper. In the present paper, using the field of linear frame [5, 4, 9], we shall consider Finslerian hypersurfaces given by a generalized β–conformal change of a Finsler metric. Our purpose is to give some relations between the original Finslerian hypersurface and the other which is Finslerian hypersurface given by generalized β–conformal change. We have also obtained that a generalized β–conformal change makes three types of hypersurfaces invariant under certain conditions.

2. Finslerian Hypersurfaces

Let Mn be an n–dimensional manifold and Fn = (Mn, L) be an n–dimensional Finsler space equipped with the fundamental function L(x, y) on Mn. The metric tensor gij(x, y) and Cartan’s C–tensor Cijk(x, y) are given by

gij=122L2yiyj,         Cijk=12gijyk,

respectively and we introduce the Cartan’s connection CΓ=(Fjki,Nji,Cjki) in Fn.

A hypersurface Mn−1 of the underlying smooth manifold Mn may be parametrically represented by the equation xi = xi(uα), where uα are Gaussian coordinates on Mn−1 and Greek indices vary from 1 to n − 1. Here, we shall assume that the matrix consisting of the projection factors Bαi=xiuα is of rank n − 1. The following notations are also employed:

Bαβi=2xiuαuβ,         B0βi=vαBαβi.

If the supporting element yi at a point (uα) of Mn−1 is assumed to be tangential to Mn−1, we may then write yi=Bαi(u)vα, i.e. vα is thought of as the supporting element of Mn−1 at the point (uα). Since the function (u, v) = L{x(u), y(u, v)} gives rise to a Finsler metric of Mn−1, we get a (n − 1)–dimensional Finsler space Fn−1 = {Mn−1, (u, v)}.

At each point (uα) of Fn−1, the unit normal vector Ni(u, v) is defined by

gijBαiNj=0,         gijNiNj=1.

If Biα, Ni is the inverse matrix of (Bαi, Ni), we have

BαiBiβ=δαβ,         BαiNi=0,         NiNi=1         and         BαiBjα+NiNj=δji.

Making use of the inverse matrix (gαβ) of (gαβ), we get

Biα=gαβgijBβj,         Ni=gijNj.

For the induced Cartan’s connection ICΓ=(Fβγα,Nαβ,Cβγα) on Fn−1, the second fundamental h–tensor Hαβ and the normal curvature Hα are respectively given by [9]

Hαβ=Ni(Bαβi+FjkiBαjBβk)+MαHβ,Hα=Ni(B0βi+NjiBβj),

where

Mα=CijkBαiNjNk.

Contracting Hαβ by vα, we immediately get H0β = Hαβvα = Hβ. Furthermore the second fundamental v–tensor Mαβ is given by [8]

Mαβ=CijkBαiBβiNk.
3. Finsler Space with Generalized β–Conformal Change

Let (Mn, L) be a Finsler space Fn, where Mn is an n–dimensional differentiable manifold equipped with a fundamental function L. A change in fundamental metric L, defined by equation (1.4), is called generalized β–conformal change, where σ(x) is conformal factor and function of position only and β1), β2), …, βm) all are linearly independent one-form and defined as βr)=bir)yi.

Homogeneity of f gives

eσLf0+frβr)=f,

where the subscripts ‘0’ and ‘r’ denote the partial derivative with respect to L and βr) respectively. The letters r, s, t, r′ and s′ vary from 1 to m throughout the paper. Summation convention is applied for the indices r, s, t, r′ and s′. If we write n = (Mn, ), then the Finsler space n is said to be obtained from Fn by generalized β–conformal change. The quantities corresponding to n are denoted by putting bar on those quantities.

To find the relation between fundamental quantities of (Mn, L) and (Mn, ), we use the following results:

˙iβr)=bir),         ˙iL=li,         ˙jli=L-1hij,

where ∂̇i stands for yi and hij are components of angular metric tensor of (Mn, L) given by

hij=gij-lilj=L˙i˙jL.

Differentiating (3.1) with respect to L and βs) respectively, we get

eσLf00+f0rβr)=0

and

eσLf0s+frsβr)=0.

The successive differentiation of (1.4) with respect to yi and yj give

l¯i=eσf0li+frbir),h¯ij=eσff0Lhij+e2σff00lilj+eσff0r(bjr)li+bir)lj)+ffrsbir)bjs).

Using equations (3.3) and (3.4) in equation (3.6), we have

h¯ij=eσff0Lhij+ffrs(bir)-βr)Lli)(bjs)-βs)Llj).

If we put mir)=bir)-βr)Lli, equation (3.7) may be written as

h¯ij=eσff0Lhij+ffrsmir)mjs).

From equations (3.5) and (3.8), we get the following relation between metric tensors of (Mn, L) and (Mn, )

g¯ij=eσff0Lgij+eσ(eσf02-ff0L)lilj+ffrsmir)mjs)+eσf0fr(bir)lj+bjr)li)+frfsbir)bjs).

Now,

(a)˙imjr)=-1L(mir)lj+βr)Lhij),(b)˙if=eσf0li+frbir),(c)˙ifrs=eσfrs0li+frstbit).

Differentiating equation (3.8) with respect to yk and using equations (3.2), (3.3), (3.4), (3.5), (3.9) and (3.10), we get

C¯ijk=p0Cijk+p1(hijmkr)+hjkmir)+hkimjr))+p2mir)mjs)mkt),

where

p0=eσff0LCijk,         p1=eσ2L(f0fr+ff0r),p2=12(frsft+fstfr+ftrfs+ffrst).
4. Hypersurfaces Given by a Generalized β–Conformal Change

Consider a Finslerian hypersurface Fn−1 = {Mn−1, (u, v)} of the Fn and another Finslerian hypersurface n−1 = {Mn−1, (u, v)} of the n given by generalized β–conformal change. Let Ni be the unit vector at each point of Fn−1 and (Biα, Ni) be the inverse matrix of (Biα, Ni). The function Biα may be considered as components of (n − 1) linearly independent tangent vectors of Fn−1 and they are invariant under generalized β–conformal change. Thus, we shall show that a unit normal vector i(u, v) of n−1 is uniquely determined by

g¯ijBαiN¯j=0,         g¯ijN¯iN¯j=1.

Contracting (3.9) by NiNj and paying attention to (2.1) and the fact that liNi = 0, we have

g¯ijNiNj=p0+p(bir)bjs)NiNj),

where p = ffrs + frfs. Therefore, we obtain

g¯ij(±Nip0+p(bir)bjs)NiNj))(±Njp0+p(bir)bjs)NiNj))=1.

Hence, we can put

N¯i=Nip0+p(bir)bjs)NiNj),

where we have chosen the positive sign in order to fix an orientation.

Using equations (3.9), (4.3) and from first condition of (4.1), we have

Bαi(2p1Lli+pbir)).bjs)Njp0+p(bir)bjs)NiNj)=0.

If Bαi(2p1Lli+pbir)=0, then contracting it by vα and using yi=Bαivα, we get L = 0 or βr) = 0 which is a contradiction with the assumption that L > 0. Hence bjs)Nj=0. Therefore equation (4.3) is written as

N¯i=Nip0.

Summarizing the above, we obtain

Proposition 4.1

For a field of linear frame (B1i,B2i,,Bn-1i, Ni) of Fn there exists a linear frame (B1i,B2i,Bn-1i,N¯i=Nip0) of F̄n such that (4.1) is satisfied along F̄n−1and thenbir)is tangential to both of the hypersurfaces Fn−1and F̄n−1.

The quantities B¯iα are uniquely defined along n−1 by

B¯iα=g¯αβg¯ijBβj

where αβ is the inverse matrix of αβ. Let (B¯iα, i) be the inverse matrix of (Bαi, i), then we have

BαiB¯iβ=δαβ,         BαiN¯i=0,         N¯iN¯i=1.

Furthermore BαiB¯jα+N¯iN¯j=δji. We also get i = ijj which in view of (3.5), (3.9) and (4.5) gives

N¯i=p0Ni.

We denote the Cartan’s connection of Fn and n by (Fjki,Nji,Cjki) and (F¯jki,N¯ji,C¯jki) respectively and put Djki=F¯jki-Fjki which will be called difference tensor. We choose the vector field br)i in Fn such that

Djki=Ajkbr)i+Bjkli+δjiDk+δkiDj,

where Ajk and Bjk are components of a symmetric covariant tensor of second order and Di are components of a covariant vector. Since Nibr)i = 0, Nili = 0 and δjiNiBαj=0, from (4.7), we get

NiDjkiBαjBβk=0         and      NiD0kiBβk=0.

Therefore, from (2.3) and (4.6), we get

H¯α=p0Hα.

If each path of a hypersurface Fn−1 with respect to the induced connection also a path of the enveloping space Fn, then Fn−1 is called a hyperplane of the first kind. A hyperplane of the first kind is characterized by Hα = 0 [8]. Hence from (4.9), we have

Theorem 4.1

Ifbir)(x)be a vector field in Fn satisfying (4.7), then a hypersurface Fn−1is a hyperplane of the first kind if and only if the hypersurface F̄n−1is a hyperplane of the first kind.

Next contracting (3.11) by BαiN¯jN¯k and paying attention to (4.5), mir)Ni=0, hjkNjNk = 1 and hijBαiNj=0, we get

M¯α=Mα+p1p0mir)Bαi.

From (2.3), (4.6), (4.8), we have

H¯αβ=p0Hαβ.

If each h–path of a hypersurface Fn−1 with respect to the induced connection is also h–path of the enveloping space Fn, then Fn−1 is called a hyperplane of the second kind. A hyperplane of the second kind is characterized by Hαβ = 0 [8]. Since Hαβ = 0 implies that Hα = 0 from (4.9) and (4.10), we have the following:

Theorem 4.2

Ifbir)(x)be a vector field in Fn satisfying (4.7), then a hypersurface Fn−1is a hyperplane of the second kind if and only if the hypersurface F̄n−1is a hyperplane of the second kind.

Finally contracting (3.11) by BαiBβjN¯k and paying attention to (4.5), we have

M¯αβ=p0Mαβ.

If the unit normal vector of Fn−1 is parallel along each curve of Fn−1, then Fn−1 is called a hyperplane of third kind. A hyperplane of the third kind is characterized by Hαβ = 0, Mαβ = 0 [8]. From (4.10) and (4.11), we have:

Theorem 4.3

Ifbir)(x)be a vector field in Fn satisfying (4.7), then a hypersurface Fn−1is a hyperplane of the third kind if and only if the hypersurface F̄n−1is a hyperplane of the third kind.

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