C-conformal Metric Transformations on Finslerian Hypersurface

The purpose of the paper is to give some relation between the originalFinslerian hypersurface and other C-conformal Finslerian hypersufaces. In this pa-per we dene three types of hypersufaces, which were called a hyperplane of the 1stkind, hyperplane of the 2nd kind and hyperplane of the 3rd kind under considerationof C-conformal metric transformation.DOI : http://dx.doi.org/10.22342/jims.17.2.2.59-66


Introduction
The conformal theory and its related concepts of Finsler spaces was initiated by Knebelman in 1929. M. Hashiguchi [1] introduced a special change called Cconformal change which satisfies C-condition. The theory of Special Finsler spaces and their properties were studied by M. Matsumoto [8], C. Shibata [13] et al and authors like H. Izumi [2], S. Kikuchi [4] et al have given the condition for Finsler space to be conformally flat. C. Shibata and H. Azuma [13] have studied C-conformal invariant tensor of Finsler metric. The author M. Kitayama ([5], [6], [7]) have studied Finsler spaces admitting a parallel vector field and also studied Finslerian hypersurface and metric transformations. The authors H.G. Nagaraja, C.S. Bagewadi and H. Izumi [9] have published a paper on infinitesimal h-conformal motions of Finsler metric.
The authors S.K. Narasimhamurthy and C.S. Bagewadi ( [10], [11]) have published a paper on C-conformal Special Finsler spaces admitting a parallel vector field and the same authors have also studied on Infinitesimal C-conformal motions of special Finsler spaces.

Preliminaries
A Finsler space, we mean a triple F n = (M, D, L), where M denotes ndimensional differentiable manifold, D is an open subset of a tangent vector bundle T M endowed with the differentiable structure induced by the differentiable manifold T M and L : D → R is a differentiable mapping having the properties ii) L(x, λy) = |λ|L(x, y), f or any (x, y) ∈ D and λ ∈ R, such that (x, λy) ∈ D, The metric tensor g ij (x, y) and Cartan's C-tensor C ijk are given by [12]: where∂ j = ∂ ∂y i and∂ i = ∂ ∂x i . We use the following [12]: The Berwald connection and the Cartan connection of F n are given by BΓ = (G i jk , N i j , 0) and CΓ = (F i jk , N i j , C i jk ) respectively.
A hypersurface M n−1 of the underlying smooth manifold M n may be parametrically represented by the equation where u α are Gaussian coordinates on M n−1 and Greek indices take values 1 to n-1. Here we shall assume that the matrix consisting of the projection factors B i α = ∂x i /∂u α is of rank (n-1). The following notations are also employed [6]: . If the supporting element y i at a point (u α ) of M n−1 is assumed to be tangential to M n−1 , we may then write Making use of the inverse (g αβ ) of (g αβ ), we get For the induced Cartan connections ICΓ = (F α βγ , N α β , C α βγ ) on F n−1 , the second fundamental h-tensor H αβ and the normal curvature tensor H α are given by Further more we have to put

C-Conformal Finsler Space
We shall consider conformal change of a Finsler metric formed by L → L = e σ(x) L, where σ is conformal factor depends on the point x only and under this change we have another Finsler space F n = (M n , L) on the same underlying manifold M n . M. Hashiguchi [1] introduced the special change named C-conformal change which is by definition, a non-homothetic conformal change satisfying where C i jk = g im (∂ j g km )/2, σ i = g im σ m , σ m = ∂σ/∂x m , σ j = g ij σ j . From (1) and by symmetry of lower indices of C ijk , we have In the following the quantity with bar will be defined in C-conformal Finsler space F n , and the quantity without bar will be defined in Finsler space F n .

Hypersurface Given by a C-Conformal Change
We now consider a Finsler hypersurface F n−1 = (M n−1 , L(u, v)) of the Finsler space F n and another Finsler hypersurface F n−1 = (M n−1 , L(u, v)) of the Finsler space F n given by the C-conformal change.
Let N i (u, v) be a unit normal vector at each point of the F n−1 , and as component of n-1 linearly independent tangent vectors of F n−1 and they are invariant under the C-conformal change. Thus we shall show that a unit normal vector By means of (2) and (6), we get Therefore we can put where we have chosen the sign '+' in order to fix an orientation. It is obvious that N i (u, v) satisfies (2), hence we obtain: The quantities B α i are uniquely defined along F n−1 by where (g αβ ) is the inverse metric of (g αβ ).
We have from (6(e)), Differentiating (9) by y j and from (6(f)), we obtain We assume that N i σ i = 0. i.e., σ i (x) is tangential to F n−1 and using the condition N i y i = 0, then we have Differentiating (10) by y j , we have where we used If each path of the hypersurface F n−1 with respect to the induced connection is also a path of the ambient space F n , then F n−1 is called a 'hyperplane of the 1 st kind'.
A hyperplane of the 1 st kind is characterized by H α = 0. From (3(ii)) and using (8), we have . Thus from (11), we obtained H α = e σ H α . Hence we state the following: Now from (6(h)), the so called difference tensor D i jk has the following form Where we use σ 0 = σ i y i and equation (5). Thus we state the following: Definition 4.2. If each h-path of a hypersurface F n−1 with respect to the induced connection is also h-path of the ambient space F n , then F n−1 is called a 'hyperplane of the 2 nd kind'. A hyperplane of the 2 nd kind is characterized by H αβ = 0. From (3(i)), we have Under the C-conformal change, (12) can be written as Using equations (12) and (13), we get Thus by virtue of lemma (4.1), therefore we state the following: Theorem 4.2. A Finsler hypersurface F n−1 is a hyperplane of the 2 nd kind if and only if the C-conformal Finsler hypersurface F n−1 is a hyperplane of the 2 nd kind, provided σ i (x) is tangential to F n−1 .
Definition 4.3. If the unit normal vector of F n−1 is parallel along each curve of F n−1 , then F n−1 is called a 'hyperplane of the 3 rd kind'. A hyperplane of the 3 rd kind is characterized by H αβ = M αβ = 0. From (4), under C-conformal change the tensor M αβ can be written as = e −σ C ijk B ij αβ N k , = e −σ M αβ .
By characterization of hyperplane of the 3 rd kind and (15), we have H αβ = M αβ = 0. Thus by virtue of lemma (4.1), we state the following: provided σ i (x) is tangential to F n−1 .