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:

where β = b_i(x) yⁱ, b_i(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

where f is any positively homogeneous function of degree one in L, β¹⁾, β²⁾, …, β^m), where β¹⁾, β²⁾, …, β^m) 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

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

where f is any positively homogeneous function of degree one in e^σL, β¹⁾, β²⁾, …, β^m).

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.

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 g_ij(x, y) and Cartan’s C–tensor C_ijk(x, y) are given by

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(u^α), where u^α 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=∂xi∂uα is of rank n − 1. The following notations are also employed:

If the supporting element yⁱ at a point (u^α) of Mⁿ⁻¹ is assumed to be tangential to Mⁿ⁻¹, we may then write yi=Bαi(u)vα, i.e. v^α is thought of as the supporting element of Mⁿ⁻¹ at the point (u^α). Since the function L̄(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, L̄(u, v)}.

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

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

Making use of the inverse matrix (g^αβ) of (g_αβ), we get

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

where

Contracting H_αβ by v^α, we immediately get H_0β = H_αβv^α = H_β. Furthermore the second fundamental v–tensor M_αβ is given by [8]

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⁾, …, β^m) all are linearly independent one-form and defined as βr)=bir)yi.

Homogeneity of f gives

where the subscripts ‘0’ and ‘r’ denote the partial derivative with respect to L and β^r) 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 F̄ⁿ = (Mⁿ, L̄), then the Finsler space F̄ⁿ is said to be obtained from Fⁿ by generalized β–conformal change. The quantities corresponding to F̄ⁿ are denoted by putting bar on those quantities.

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

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

Differentiating (3.1) with respect to L and β^s) respectively, we get

and

The successive differentiation of (1.4) with respect to yⁱ and y^j give

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

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

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

Now,

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

where

Consider a Finslerian hypersurface Fⁿ⁻¹ = {Mⁿ⁻¹, L̄(u, v)} of the Fⁿ and another Finslerian hypersurface F̄ⁿ⁻¹ = {Mⁿ⁻¹, L̄(u, v)} of the F̄ⁿ given by generalized β–conformal change. Let Nⁱ be the unit vector at each point of Fⁿ⁻¹ and (Biα, N_i) 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 N̄ⁱ(u, v) of F̄ⁿ⁻¹ is uniquely determined by

Contracting (3.9) by NⁱN^j and paying attention to (2.1) and the fact that l_iNⁱ = 0, we have

where p = ff_rs + f_rf_s. Therefore, we obtain

Hence, we can put

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

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

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 F̄ⁿ⁻¹ by

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

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

We denote the Cartan’s connection of Fⁿ and F̄ⁿ 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 b^r)i in Fⁿ such that

where A_jk and B_jk are components of a symmetric covariant tensor of second order and D_i are components of a covariant vector. Since N_ib^r)i = 0, N_ilⁱ = 0 and δjiNiBαj=0, from (4.7), we get

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

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 H_α = 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, h_jkN^jN^k = 1 and hijBαiNj=0, we get

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

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 H_αβ = 0 [8]. Since H_αβ = 0 implies that H_α = 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

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 H_αβ = 0, M_αβ = 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.