On a Hypersurface of a Finsler Space with Randers Change of Matsumoto Metric

Abstract

The present paper contains certain geometrical properties of a hypersurface of a Finsler space with Randers change of Matsumoto metric.

1. Introduction

The concept of Finsler spaces with (𝛼,𝛽)-metric was introduced by Matsumoto [1]. It was later discussed by several authors such as Shibata and others [2–4]. It has plenty of applications in various fields such as physics, mechanics, seismology, biology, and ecology [5–8]. Matsumoto introduced a special type of (𝛼,𝛽)-metric of the form 𝐿=𝛼²/(𝛼−𝛽), 𝛼=𝑎_𝑖𝑗(𝑥)𝑦^𝑖𝑦^𝑗, and 𝛽=𝑏_𝑖(𝑥)𝑦^𝑖, which is slope-of-a-mountain metric and is known as Matsumoto metric [9]. This metric has enriched Finsler geometry and it has provided researchers an important tool to work with significantly in this field [7, 10].

A change of Finsler metric 𝐿(𝑥,𝑦)→𝐿(𝑥,𝑦)=𝐿(𝑥,𝑦)+𝑏_𝑖(𝑥)𝑦^𝑖 is called Randers change of metric. The notion of a Randers change was proposed by Matsumoto, named by Hashiguchi and Ichijyo [11] and studied in detail by Shibata [12]. A Randers change of Matsumoto metric is given by 𝐿(𝑥,𝑦)=𝛼²/(𝛼−𝛽)+𝛽. Recently, Nagaraja and Kumar [13] studied the properties of a Finsler space with the Randers change of Matsumoto metric.

Matsumoto [14] presented the theory of Finslerian hypersurface. The present authors (Gupta and Pandey [15, 16]) obtained certain geometrical properties of hypersurfaces of some special Finsler spaces. Singh and Kumari [17] discussed a hypersurface of a Finsler space with Matsumoto metric.

In this paper, we consider an 𝑛-dimensional Finsler space 𝐹^𝑛=(𝑀^𝑛,𝐿) with the Randers change of Matsumoto metric 𝐿=𝛼²/(𝛼−𝛽)+𝛽 and find certain geometrical properties of a hypersurface of the Finsler space with above metric. The paper is organized as follows.

Section 2 consists of Preliminaries relevant to the subsequent sections. The induced Cartan connection for hypersurface of a Finsler space is defined in Section 3. Necessary and sufficient conditions under which the hypersurface of the above Finsler space is a hyperplane of first, second and third kind are obtained in Section 4.

2. Preliminaries

Let 𝑀^𝑛 be an 𝑛-dimensional smooth manifold and let 𝐹^𝑛=(𝑀^𝑛,𝐿) be an 𝑛-dimensional Finsler space equipped with Randers change of Matsumoto metric function 𝛼𝐿=²𝛼−𝛽+𝛽. (1) The derivative of above Randers change of Matsumoto metric with respect to 𝛼 and 𝛽 is given by 𝐿_𝛼=𝛼²−2𝛼𝛽𝛼−𝛽²,𝐿_𝛽=2𝛼²+𝛽²−2𝛼𝛽𝛼−𝛽²,𝐿_𝛼𝛼=2𝛽²𝛼−𝛽³,𝐿_𝛽𝛽=2𝛼²𝛼−𝛽³,𝐿_𝛼𝛽=−2𝛼𝛽𝛼−𝛽³, (2) where 𝐿_𝛼=𝜕𝐿𝜕𝛼,𝐿_𝛽=𝜕𝐿𝜕𝛽,𝐿_𝛼𝛼=𝜕𝐿_𝛼,𝐿𝜕𝛼_𝛽𝛽=𝜕𝐿_𝛽𝜕𝛽,𝐿_𝛼𝛽=𝜕𝐿_𝛼.𝜕𝛽 (3) The normalized element of support 𝑙_𝑖=𝜕̇_𝑖𝐿 is given by 𝑙_𝑖=𝛼⁻¹𝐿_𝛼𝑦_𝑖+𝐿_𝛽𝑏_𝑖, (4) where 𝑦_𝑖=𝑎_𝑖𝑗𝑦^𝑗. The angular metric tensor ℎ_𝑖𝑗=𝐿𝜕̇_𝑖𝜕̇_𝑗𝐿 is given by ℎ_𝑖𝑗=𝑝𝑎_𝑖𝑗+𝑞₀𝑏_𝑖𝑏_𝑗+𝑞₁𝑏_𝑖𝑦_𝑗+𝑏_𝑗𝑦_𝑖+𝑞₂𝑦_𝑖𝑦_𝑗, (5) where 𝑝=𝐿𝐿_𝛼𝛼⁻¹=𝛼³−𝛼²𝛽+2𝛽³−3𝛼𝛽²𝛼−𝛽³,𝑞₀=𝐿𝐿_𝛽𝛽=2𝛼²𝛼²+𝛼𝛽−𝛽²𝛼−𝛽⁴,𝑞₁=𝐿𝐿_𝛼𝛽𝛼⁻¹𝛼=−2𝛽²+𝛼𝛽−𝛽²𝛼−𝛽⁴,𝑞₂=𝐿𝛼⁻²𝐿_𝛼𝛼−𝐿_𝛼𝛼⁻¹=𝛼²+𝛼𝛽−𝛽²3𝛽−𝛼𝛼𝛼−𝛽⁴. (6) The fundamental metric tensor 𝑔_𝑖𝑗=(1/2)𝜕̇_𝑖𝜕̇_𝑗𝐿² is given by 𝑔_𝑖𝑗=𝑝𝑎_𝑖𝑗+𝑝₀𝑏_𝑖𝑏_𝑗+𝑝₁𝑏_𝑖𝑦_𝑗+𝑏_𝑗𝑦_𝑖+𝑝₂𝑦_𝑖𝑦_𝑗, (7) where 𝑝₀=𝑞₀+𝐿²_𝛽=6𝛼⁴−6𝛼³𝛽+6𝛼²𝛽²−4𝛼𝛽³+𝛽⁴𝛼−𝛽⁴,𝑝₁=𝑞₁+𝐿⁻¹𝑝𝐿_𝛽=𝛼2𝛼²+3𝛽²−8𝛼𝛽𝛼−𝛽⁴,𝑝₂=𝑞₂+𝑝²𝐿⁻²=𝛽8𝛼𝛽−2𝛼²−3𝛽²𝛼𝛼−𝛽⁴. (8) Moreover, the reciprocal tensor 𝑔^𝑖𝑗 of 𝑔_𝑖𝑗 is given by 𝑔^𝑖𝑗=𝑝⁻¹𝑎^𝑖𝑗−𝑠₀𝑏^𝑖𝑏^𝑗−𝑠₁𝑏^𝑖𝑦^𝑗+𝑏^𝑗𝑦^𝑖−𝑠₂𝑦^𝑖𝑦^𝑗, (9) where 𝑏^𝑖=𝑎^𝑖𝑗𝑏_𝑗,𝑠₀=𝑝𝑝₀+𝑝₀𝑝₂−𝑝²₁𝛼²,𝑠𝜁𝑝₁=𝑝𝑝₁−𝑝₀𝑝₂−𝑝²₁𝛽,𝑠𝜁𝑝₂=𝑝𝑝₂+𝑝₀𝑝₂−𝑝²₁𝑏²𝜁𝑝,𝑏²=𝑎_𝑖𝑗𝑏^𝑖𝑏^𝑗,𝜁=𝑝𝑝+𝑝₀𝑏²+𝑝₁𝛽+𝑝₀𝑝₂−𝑝²₁𝛼²𝑏²−𝛽². (10) The Cartan tensor 𝐶_𝑖𝑗𝑘=(1/2)𝜕̇_𝑘𝑔_𝑖𝑗 is given by 2𝑝𝐶_𝑖𝑗𝑘=𝑝₁ℎ_𝑖𝑗𝑚_𝑘+ℎ_𝑗𝑘𝑚_𝑖+ℎ_𝑘𝑖𝑚_𝑗+𝛾₁𝑚_𝑖𝑚_𝑗𝑚_𝑘, (11) where 𝛾₁=𝑝𝜕𝑝₀𝜕𝛽−3𝑝₁𝑞₀,𝑚_𝑖=𝑏_𝑖−𝛼⁻²𝛽𝑦_𝑖. (12)

Let {^𝑖_𝑗𝑘} be the components of Christoffel symbols of the associated Riemannian space 𝑅^𝑛 and let ∇_𝑘 be the covariant differentiation with respect to 𝑥^𝑘 relative to this Christoffel symbols. We will use the following tensors: 2𝐸_𝑖𝑗=𝑏_𝑖𝑗+𝑏_𝑗𝑖,2𝐹_𝑖𝑗=𝑏_𝑖𝑗−𝑏_𝑗𝑖, (13) where 𝑏_𝑖𝑗=∇_𝑗𝑏_𝑖.

If we denote the Cartan connection in 𝐹^𝑛 as 𝐶Γ=(𝐹^𝑖_𝑗𝑘,𝐺^𝑖_𝑗,𝐶^𝑖_𝑗𝑘), then the difference tensor 𝐷^𝑖_𝑗𝑘=𝐹^𝑖_𝑗𝑘−{^𝑖_𝑗𝑘} of the Finsler space 𝐹^𝑛 is given by 𝐷^𝑖_𝑗𝑘=𝐵^𝑖𝐸_𝑗𝑘+𝐹^𝑖_𝑘𝐵_𝑗+𝐹^𝑖_𝑗𝐵_𝑘+𝐵^𝑖_𝑗𝑏_0𝑘+𝐵^𝑖_𝑘𝑏_0𝑗−𝑏_0𝑚𝑔^𝑖𝑚𝐵_𝑗𝑘−𝐶^𝑖_𝑗𝑚𝐴^𝑚_𝑘−𝐶^𝑖_𝑘𝑚𝐴^𝑚_𝑗+𝐶_𝑗𝑘𝑚𝐴^𝑚_𝑠𝑔^𝑖𝑠+𝜆^𝑠𝐶^𝑖_𝑗𝑚𝐶^𝑚_𝑠𝑘+𝐶^𝑖_𝑘𝑚𝐶^𝑚_𝑠𝑗−𝐶^𝑚_𝑗𝑘𝐶^𝑖_𝑚𝑠, (14) where 𝐵_𝑘=𝑝₀𝑏_𝑘+𝑝₁𝑦_𝑘,𝐵^𝑖=𝑔^𝑖𝑗𝐵_𝑗,𝐹^𝑘_𝑖=𝑔^𝑘𝑗𝐹_𝑗𝑖,𝐵_𝑖𝑗=𝑝₁𝑎_𝑖𝑗−𝛼⁻²𝑦_𝑖𝑦_𝑗+𝜕𝑝₀/𝜕𝛽𝑚_𝑖𝑚_𝑗2,𝐴^𝑚_𝑘=𝐵^𝑚_𝑘𝐸₀₀+𝐵^𝑚𝐸_𝑘0+𝐵_𝑘𝐹^𝑚₀+𝐵₀𝐹^𝑚_𝑘,𝐵^𝑘_𝑖=𝑔^𝑘𝑗𝐵_𝑗𝑖,𝜆^𝑚=𝐵^𝑚𝐸₀₀+2𝐵₀𝐹^𝑚₀. (15) The suffix “0” denotes the transvection by the supporting element 𝑦^𝑖 except for the quantities 𝑝₀, 𝑞₀, and 𝑠₀.

3. Induced Cartan Connection

A hypersurface 𝑀^𝑛−1 of the underlying manifold 𝑀^𝑛 may be represented parametrically by 𝑥^𝑖=𝑥^𝑖(𝑢^𝛼), where 𝑢^𝛼 are the Gaussian coordinates on 𝑀^𝑛−1 (Latin indices run from 1 to 𝑛, while Greek indices take values from 1 to 𝑛−1). We assume that the matrix of projection factors 𝐵^𝑖_𝛼=𝜕𝑥^𝑖/𝜕𝑢^𝛼 is of rank 𝑛−1. If the supporting element 𝑦^𝑖 at a point 𝑢=(𝑢^𝛼) of 𝑀^𝑛−1 is assumed to be tangent to 𝑀^𝑛−1, we may then write 𝑦^𝑖=𝐵^𝑖_𝛼(𝑢)𝑣^𝛼 so that 𝑣=(𝑣^𝛼) is thought of as the supporting element of 𝑀^𝑛−1 at the point 𝑢^𝛼. Since the function 𝐿(𝑢,𝑣)=𝐿(𝑥(𝑢),𝑦(𝑢,𝑣)) gives rise to a Finsler metric on 𝑀^𝑛−1, we get an (𝑛−1)-dimensional Finsler space 𝐹^𝑛−1=(𝑀^𝑛−1,𝐿(𝑢,𝑣)). The metric tensor 𝑔_𝛼𝛽 and the Cartan tensor 𝐶_𝛼𝛽𝛾 are given by 𝑔_𝛼𝛽=𝑔_𝑖𝑗𝐵^𝑖_𝛼𝐵^𝑗_𝛽,𝐶_𝛼𝛽𝛾=𝐶_𝑖𝑗𝑘𝐵^𝑖_𝛼𝐵^𝑗_𝛽𝐵^𝑘_𝛾. (16) At each point 𝑢^𝛼 of 𝐹^𝑛−1, a unit normal vector 𝑁^𝑖(𝑢,𝑣) is defined by 𝑔_𝑖𝑗𝐵^𝑖_𝛼𝑁^𝑗=0,𝑔_𝑖𝑗𝑁^𝑖𝑁^𝑗=1. (17) For the angular metric tensor ℎ_𝑖𝑗, we have ℎ_𝛼𝛽=ℎ_𝑖𝑗𝐵^𝑖_𝛼𝐵^𝑗_𝛽,ℎ_𝑖𝑗𝐵^𝑖_𝛼𝑁^𝑗=0,ℎ_𝑖𝑗𝑁^𝑖𝑁^𝑗=1. (18) The inverse projection factors 𝐵^𝛼_𝑖(𝑢,𝑣) of 𝐵^𝑖_𝛼 are defined as 𝐵^𝛼_𝑖=𝑔^𝛼𝛽𝑔_𝑖𝑗𝐵^𝑗_𝛽, (19) where 𝑔^𝛼𝛽 is the inverse of the metric tensor 𝑔_𝛼𝛽 of 𝐹^𝑛−1.

From (17) and (19), it follows that 𝐵^𝑖_𝛼𝐵^𝛽_𝑖=𝛿^𝛽_𝛼,𝐵^𝑖_𝛼𝑁_𝑖=0,𝑁^𝑖𝐵^𝛼_𝑖=0,𝑁^𝑖𝑁_𝑖=1, (20) and further 𝐵^𝑖_𝛼𝐵^𝛼_𝑗+𝑁^𝑖𝑁_𝑗=𝛿^𝑖_𝑗. (21) For the induced Cartan connection 𝐼𝐶Γ=(𝐹^𝛼_𝛽𝛾,𝐺^𝛼_𝛽,𝐶^𝛼_𝛽𝛾) on 𝐹^𝑛−1, the second fundamental ℎ-tensor 𝐻_𝛼𝛽 and the normal curvature vector 𝐻_𝛼 are given by 𝐻_𝛼𝛽=𝑁_𝑖𝐵^𝑖_𝛼𝛽+𝐹^𝑖_𝑗𝑘𝐵^𝑗_𝛼𝐵^𝑘_𝛽+𝑀_𝛼𝐻_𝛽,𝐻_𝛼=𝑁_𝑖𝐵^𝑖_0𝛼+𝐺^𝑖_𝑗𝐵^𝑗_𝛼, (22) where 𝑀_𝛼=𝐶_𝑖𝑗𝑘𝐵^𝑖_𝛼𝑁^𝑗𝑁^𝑘, 𝐵^𝑖_𝛼𝛽=𝜕²𝑥^𝑖/𝜕𝑢^𝛼𝜕𝑢^𝛽, and 𝐵^𝑖_0𝛼=𝐵^𝑖_𝛽𝛼𝑣^𝛽. It is clear that 𝐻_𝛼𝛽 is not symmetric and 𝐻_𝛼𝛽−𝐻_𝛽𝛼=𝑀_𝛼𝐻_𝛽−𝑀_𝛽𝐻_𝛼. (23) Equation (22) yields 𝐻_0𝛼=𝐻_𝛽𝛼𝑣^𝛽=𝐻_𝛼,𝐻_𝛼0=𝐻_𝛼𝛽𝑣^𝛽=𝐻_𝛼+𝑀_𝛼𝐻₀. (24) The second fundamental 𝑣-tensor 𝑀_𝛼𝛽 is defined as: 𝑀_𝛼𝛽=𝐶_𝑖𝑗𝑘𝐵^𝑖_𝛼𝐵^𝑗_𝛽𝑁^𝑘. (25) The relative ℎ- and 𝑣-covariant derivatives of 𝐵^𝑖_𝛼 and 𝑁^𝑖 are given by 𝐵^𝑖_𝛼|𝛽=𝐻_𝛼𝛽𝑁^𝑖,𝐵^𝑖_𝛼|_𝛽=𝑀_𝛼𝛽𝑁^𝑖,𝑁^𝑖_|𝛽=−𝐻_𝛼𝛽𝐵^𝛼_𝑗𝑔^𝑖𝑗,𝑁^𝑖|_𝛽=−𝑀_𝛼𝛽𝐵^𝛼_𝑗𝑔^𝑖𝑗. (26) Let 𝑋_𝑖(𝑥,𝑦) be a vector field of 𝐹^𝑛. The relative ℎ- and 𝑣-covariant derivatives of 𝑋_𝑖 are given by 𝑋_𝑖|𝛽=𝑋_𝑖|𝑗𝐵^𝑗_𝛽+𝑋_𝑖|_𝑗𝑁^𝑗𝐻_𝛽,𝑋_𝑖|_𝛽=𝑋_𝑖|_𝑗𝐵^𝑗_𝛽. (27)

Matsumoto [14] defined different kinds of hyperplanes and obtained their characteristic conditions, which are given in the following lemmas.

Lemma 1. A hypersurface 𝐹^𝑛−1 is a hyperplane of the first kind if and only if 𝐻_𝛼=0 or equivalently 𝐻₀=0.

Lemma 2. A hypersurface 𝐹^𝑛−1 is a hyperplane of the second kind if and only if 𝐻_𝛼𝛽=0.

Lemma 3. A hypersurface 𝐹^𝑛−1 is a hyperplane of the third kind if and only if 𝐻_𝛼𝛽=0=𝑀_𝛼𝛽.

4. Hypersurface 𝐹^𝑛−1(𝑐) of the Finsler Space with Randers Change of Matsumoto Metric

Let us consider the Randers change of Matsumoto metric 𝐿=𝛼²/(𝛼−𝛽)+𝛽 with the gradient 𝑏_𝑖(𝑥)=𝜕_𝑖𝑏 for a scalar function 𝑏(𝑥) and a hypersurface 𝐹^𝑛−1(𝑐) given by the equation 𝑏(𝑥)=𝑐 (constant). From parametric equation 𝑥^𝑖=𝑥^𝑖(𝑢^𝛼) of 𝐹^𝑛−1(𝑐), we get 𝜕_𝛼𝑏(𝑥(𝑢))=0=𝑏_𝑖𝐵^𝑖_𝛼, so that 𝑏_𝑖(𝑥) are regarded as covariant components of a normal vector field of 𝐹^𝑛−1(𝑐). Therefore, along 𝐹^𝑛−1(𝑐), we have 𝑏_𝑖𝐵^𝑖_𝛼=0,𝑏_𝑖𝑦^𝑖=0. (28) Therefore the induced metric 𝐿(𝑢,𝑣) of 𝐹^𝑛−1(𝑐) is given by 𝐿(𝑢,𝑣)=𝑎_𝛼𝛽𝑣^𝛼𝑣^𝛽,𝑎_𝛼𝛽=𝑎_𝑖𝑗𝐵^𝑖_𝛼𝐵^𝑗_𝛽, (29) which is a Riemannian metric.

At a point of 𝐹^𝑛−1(𝑐), from (6), (8), and (10), we have 𝑝=1,𝑞₀=2,𝑞₁=0,𝑞₂1=−𝛼²,𝑝₀=6,𝑝₁=2𝛼,𝑝₂=0,𝜁=1+2𝑏²,𝑠₀=21+2𝑏²,𝑠₁=2𝛼1+2𝑏²,𝑠₂=−4𝑏²𝛼²1+2𝑏². (30) Therefore (9) gives 𝑔^𝑖𝑗=𝑎^𝑖𝑗−21+2𝑏²𝑏^𝑖𝑏^𝑗−2𝛼1+2𝑏²×𝑏^𝑖𝑦^𝑗+𝑏^𝑗𝑦^𝑖+4𝑏²𝛼²1+2𝑏²𝑦^𝑖𝑦^𝑗, (31) using (28) we get 𝑔^𝑖𝑗𝑏_𝑖𝑏_𝑗=𝑏²1+2𝑏², (32) which gives 𝑏_𝑖(𝑥(𝑢))=𝑏²1+2𝑏²𝑁_𝑖, (33) where 𝑏 is the length of the vector 𝑏^𝑖. Using (31) and (33) we get 𝑏^𝑖=𝑎^𝑖𝑗𝑏_𝑗=𝑏²1+2𝑏²𝑁^𝑖+2𝑏²𝛼⁻¹𝑦^𝑖. (34)

Theorem 4. Let 𝐹^𝑛 be a Finsler space with Randers change of Matsumoto metric 𝐿=𝛼²/(𝛼−𝛽)+𝛽 with a gradient 𝑏_𝑖(𝑥)=𝜕_𝑖𝑏(𝑥) and let 𝐹^𝑛−1(𝑐) be a hypersurface of 𝐹^𝑛, which is given by 𝑏(𝑥)=𝑐 (constant). Then the induced metric on 𝐹^𝑛−1(𝑐) is Riemannian metric given by (29), and the scalar function 𝑏(𝑥) is given by (33) and (34).

Along 𝐹^𝑛−1(𝑐), the angular metric tensor and the metric tensor of 𝐹^𝑛 are given by ℎ_𝑖𝑗=𝑎_𝑖𝑗+2𝑏_𝑖𝑏_𝑗−𝑦_𝑖𝑦_𝑗𝛼²,𝑔_𝑖𝑗=𝑎_𝑖𝑗+6𝑏_𝑖𝑏_𝑗+2𝛼𝑏_𝑖𝑦_𝑗+𝑏_𝑗𝑦_𝑖. (35)(36) If ℎ^(𝑎)_𝛼𝛽 denote the angular metric tensor of the Riemannian metric 𝑎_𝑖𝑗(𝑥), then, using (28), (35), and (18), we get ℎ_𝛼𝛽=ℎ^(𝑎)_𝛼𝛽. (37) From (8), we get 𝜕𝑝₀=𝜕𝛽18𝛼⁴−6𝛼³𝛽𝛼−𝛽⁵. (38) Thus, along 𝐹^𝑛−1(𝑐), 𝜕𝑝₀/𝜕𝛽=18/𝛼, and therefore (12) gives 𝛾₁=6/𝛼,𝑚_𝑖=𝑏_𝑖. Then the Cartan tensor becomes 𝐶_𝑖𝑗𝑘=1𝛼ℎ_𝑖𝑗𝑏_𝑘+ℎ_𝑗𝑘𝑏_𝑖+ℎ_𝑘𝑖𝑏_𝑗+3𝛼𝑏_𝑖𝑏_𝑗𝑏_𝑘, (39) and therefore, using (18), (25), and (28), we get 𝑀_𝛼𝛽=1𝛼𝑏²1+2𝑏²ℎ_𝛼𝛽,𝑀_𝛼=0, (40) and hence from (23) it follows that 𝐻_𝛼𝛽 is symmetric. Thus we have the following.

Theorem 5. The second fundamental v-tensor of Finsler hypersurface 𝐹^𝑛−1(𝑐) of Finsler space with Randers change of Matsumoto metric, is given by (40), and the second fundamental ℎ-tensor is symmetric.

Taking ℎ-covariant derivative of (28) with respect to the induced connection, we get 𝑏_𝑖|𝛽𝐵^𝑖_𝛼+𝑏_𝑖𝐵^𝑖_𝛼|𝛽=0. (41) Applying (27) for the vector 𝑏_𝑖, we get 𝑏_𝑖|𝛽=𝑏_𝑖|𝑗𝐵^𝑗_𝛽+𝑏_𝑖|_𝑗𝑁^𝑗𝐻_𝛽. (42) Using this and 𝐵^𝑖_𝛼|𝛽=𝐻_𝛼𝛽𝑁^𝑖, (41) becomes 𝑏_𝑖|𝑗𝐵^𝑖_𝛼𝐵^𝑗_𝛽+𝑏_𝑖|_𝑗𝐵^𝑖_𝛼𝑁^𝑗𝐻_𝛽+𝑏_𝑖𝐻_𝛼𝛽𝑁^𝑖=0. (43) Since 𝑏_𝑖|_𝑗=−𝑏_ℎ𝐶^ℎ_𝑖𝑗, using (33) and (40), we get 𝑏_𝑖|_𝑗𝐵^𝑖_𝛼𝑁^𝑗=−𝑏²1+2𝑏²𝑀_𝛼=0. (44) Thus (43) gives 𝑏²1+2𝑏²𝐻_𝛼𝛽+𝑏_𝑖|𝑗𝐵^𝑖_𝛼𝐵^𝑗_𝛽=0. (45) Since 𝐻_𝛼𝛽 is symmetric, it is clear that 𝑏_𝑖|𝑗 is symmetric. Further contracting (45) with 𝑣^𝛽 and then with 𝑣^𝛼, we get 𝑏²1+2𝑏²𝐻_𝛼+𝑏_𝑖|𝑗𝐵^𝑖_𝛼𝑦^𝑗=0,𝑏²1+2𝑏²𝐻₀+𝑏_𝑖|𝑗𝑦^𝑖𝑦^𝑗=0. (46) In view of Lemma 1, the hypersurface 𝐹^𝑛−1(𝑐) is a hyperplane of the first kind if and only if 𝑏_𝑖|𝑗𝑦^𝑖𝑦^𝑗=0. Here 𝑏_𝑖|𝑗 being the covariant derivative with respect to the Cartan connection of 𝐹^𝑛 may depend on 𝑦^𝑖.

Since 𝑏_𝑖 is a gradient vector, from (13), we have 𝐸_𝑖𝑗=𝑏_𝑖𝑗,𝐹_𝑖𝑗=0. Thus (14) reduces to 𝐷^𝑖_𝑗𝑘=𝐵^𝑖𝑏_𝑗𝑘+𝐵^𝑖_𝑗𝑏_0𝑘+𝐵^𝑖_𝑘𝑏_0𝑗−𝑏_0𝑚𝑔^𝑖𝑚𝐵_𝑗𝑘−𝐶^𝑖_𝑗𝑚𝐴^𝑚_𝑘−𝐶^𝑖_𝑘𝑚𝐴^𝑚_𝑗+𝐶_𝑗𝑘𝑚𝐴^𝑚_𝑠𝑔^𝑖𝑠+𝜆^𝑠𝐶^𝑖_𝑗𝑚𝐶^𝑚_𝑠𝑘+𝐶^𝑖_𝑘𝑚𝐶^𝑚_𝑠𝑗−𝐶^𝑚_𝑗𝑘𝐶^𝑖_𝑚𝑠. (47) In view of (30) and (31), the relations in (15) become to 𝐵_𝑖=6𝑏_𝑖+2𝛼𝑦_𝑖,𝐵^𝑖=21+2𝑏²𝑏^𝑖+2𝛼1+2𝑏²𝑦^𝑖,𝐵_𝑖𝑗=1𝛼𝑎_𝑖𝑗−1𝛼³𝑦_𝑖𝑦_𝑗+9𝛼𝑏_𝑖𝑏_𝑗,𝐵^𝑖_𝑗=𝑔^𝑘𝑖𝐵_𝑘𝑗,𝐴^𝑚_𝑘=𝐵^𝑚_𝑘𝑏₀₀+𝐵^𝑚𝑏_𝑘0,𝜆^𝑚=𝐵^𝑚𝑏₀₀. (48) By virtue of (28) we have 𝐵_𝑖0=0, 𝐵^𝑖₀=0which leads 𝐴^𝑚₀=𝐵^𝑚𝑏₀₀. Therefore we have 𝐷^𝑖_𝑗0=𝐵^𝑖𝑏_𝑗0+𝐵^𝑖_𝑗𝑏₀₀−𝐵^𝑚𝐶^𝑖_𝑗𝑚𝑏₀₀,𝐷^𝑖₀₀=𝐵^𝑖𝑏₀₀=2𝑏^𝑖1+2𝑏²+2𝑦^𝑖𝛼1+2𝑏²𝑏₀₀. (49) Using the relation (28), we get 𝑏_𝑖𝐷^𝑖_𝑗0=2𝑏²1+2𝑏²𝑏_𝑗0+1+9𝑏²𝛼1+2𝑏²𝑏₀₀𝑏_𝑗−21+2𝑏²𝑏^𝑚𝑏_𝑖𝐶^𝑖_𝑗𝑚𝑏₀₀,𝑏_𝑖𝐷^𝑖₀₀=2𝑏²1+2𝑏²𝑏₀₀. (50)(51)𝑏_𝑖|𝑗 is the covariant derivative of 𝑏_𝑖 with respect to 𝑥^𝑗 relative to the Cartan connection of 𝐹^𝑛, and 𝑏_𝑖𝑗=∇_𝑗𝑏_𝑖 is the covariant derivative of 𝑏_𝑖 with respect to 𝑥^𝑗 relative to the Riemannian connection: 𝑏_𝑖|𝑗−𝑏_𝑖𝑗=𝜕_𝑗𝑏_𝑖−𝐹^𝑟_𝑖𝑗𝑏_𝑟−𝜕_𝑗𝑏_𝑖−^𝑟_𝑖𝑗𝑏_𝑟=−𝐹^𝑟_𝑖𝑗−^𝑟_𝑖𝑗𝑏_𝑟=−𝐷^𝑟_𝑖𝑗𝑏_𝑟,thatis,𝑏_𝑖|𝑗=𝑏_𝑖𝑗−𝐷^𝑟_𝑖𝑗𝑏_𝑟. (52) Using (51), we get 𝑏_𝑖|𝑗𝑦^𝑖𝑦^𝑗=𝑏₀₀−𝐷^𝑟₀₀𝑏_𝑟=11+2𝑏²𝑏₀₀. (53) Consequently, (46) may be written as 𝑏²1+2𝑏²𝐻₀+11+2𝑏²𝑏₀₀=0. (54) Thus the condition 𝐻₀=0 is equivalent to 𝑏₀₀=0, where 𝑏_𝑖𝑗 does not depend upon 𝑦^𝑖. Since 𝑦^𝑖 is to satisfy (28), the condition is written as 𝑏_𝑖𝑗𝑦^𝑖𝑦^𝑗=(𝑏_𝑖𝑦^𝑖)(𝑐_𝑗𝑦^𝑗) for some 𝑐_𝑗(𝑥), so that we have 2𝑏_𝑖𝑗=𝑏_𝑖𝑐_𝑗+𝑏_𝑗𝑐_𝑖. (55) Using (28), it follows that 𝑏₀₀=0,𝑏_𝑖𝑗𝐵^𝑖_𝛼𝐵^𝑗_𝛽=0,𝑏_𝑖𝑗𝐵^𝑖_𝛼𝑦^𝑗=0. (56) Again (48) and (55) gives 𝑏_𝑖0𝑏^𝑖=𝑐₀𝑏²2,𝜆^𝑚=0,𝐴^𝑖_𝑗𝐵^𝑗_𝛽𝐵=0,_𝑖𝑗𝐵^𝑖_𝛼𝐵^𝑗_𝛽=1𝛼ℎ_𝛼𝛽. (57) Using the (25), (31), (34), (40), and (47), we get 𝑏_𝑟𝐷^𝑟_𝑖𝑗𝐵^𝑖_𝛼𝐵^𝑗_𝛽𝑐=−⁰𝑏²2𝛼1+2𝑏²²ℎ_𝛼𝛽. (58) Therefore in view of (52), (45) reduces to 𝑏²1+2𝑏²𝐻_𝛼𝛽+𝑐₀𝑏²2𝛼1+2𝑏²²ℎ_𝛼𝛽=0. (59)

Theorem 6. The necessary and sufficient condition for the hypersurface 𝐹^𝑛−1(𝑐) of Finsler space with Randers change of Matsumoto metric to be hyperplane of the first kind is (55) and in this case the second fundamental ℎ-tensor of hypersurface 𝐹^𝑛−1(𝑐) is proportional to its angular metric tensor.

In view of Lemma 2, the hypersurface 𝐹^𝑛−1(𝑐) is a hyperplane of the second kind if and only if 𝐻_𝛼𝛽=0. Thus from (59) we get 𝑐₀=𝑐_𝑖(𝑥)𝑦^𝑖=0. Therefore there exists a function 𝑒(𝑥) such that 𝑐_𝑖(𝑥)=𝑒(𝑥)𝑏_𝑖(𝑥). Thus (55) gives 𝑏_𝑖𝑗=𝑒𝑏_𝑖𝑏_𝑗. (60)

Theorem 7. The necessary and sufficient condition for the hypersurface 𝐹^𝑛−1(𝑐) of Finsler space with Randers change of Matsumoto metric, to be hyperplane of the second kind is (60).

In view of (40) and Lemma 3, we have the following.

Theorem 8. The hypersurface 𝐹^𝑛−1(𝑐) of Finsler space with Randers change of Matsumoto metric is not a hyperplane of the third kind.

Acknowledgment

The authors are thankful to the referee for his/her valuable suggestions.