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

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) = α/(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

A change of Finsler metric (, ) → (, ) = (, ) +   ()  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 (, ) =  2 /( − ) + .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 -dimensional Finsler space   = (  , ) with the Randers change of Matsumoto metric  =  2 /( − ) +  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.

Preliminaries
Let   be an -dimensional smooth manifold and let   = (  , ) be an -dimensional Finsler space equipped with Randers change of Matsumoto metric function The derivative of above Randers change of Matsumoto metric with respect to  and  is given by where The normalized element of support   = ∂   is given by where   =     .The angular metric tensor ℎ  =  ∂  ∂   is given by where The fundamental metric tensor   = (1/2) ∂  ∂   2 is given by where Moreover, the reciprocal tensor   of   is given by where The Cartan tensor   = (1/2) ∂    is given by where Let {   } be the components of Christoffel symbols of the associated Riemannian space   and let ∇  be the covariant differentiation with respect to   relative to this Christoffel symbols.We will use the following tensors: where   = ∇    .
If we denote the Cartan connection in   as Γ = (   ,    ,    ), then the difference tensor    =    − {   } of the Finsler space   is given by where The suffix "0" denotes the transvection by the supporting element   except for the quantities  0 ,  0 , and  0 .Geometry 3

Induced Cartan Connection
A hypersurface  −1 of the underlying manifold   may be represented parametrically by   =   (  ), where   are the Gaussian coordinates on  −1 (Latin indices run from 1 to , while Greek indices take values from 1 to  − 1).We assume that the matrix of projection factors    =   /  is of rank  − 1.If the supporting element   at a point  = (  ) of  −1 is assumed to be tangent to  −1 , we may then write   =    ()V  so that V = (V  ) is thought of as the supporting element of  −1 at the point   .Since the function  (, V) = ((), (, V)) gives rise to a Finsler metric on  −1 , we get an ( − 1)-dimensional Finsler space  −1 = ( −1 ,  (, V)).The metric tensor   and the Cartan tensor   are given by At each point   of  −1 , a unit normal vector   (, V) is defined by For the angular metric tensor ℎ  , we have The inverse projection factors    (, V) of    are defined as where   is the inverse of the metric tensor   of  −1 .From ( 17) and ( 19), it follows that and further For the induced Cartan connection Γ = (   ,    ,    ) on  −1 , the second fundamental ℎ-tensor   and the normal curvature vector   are given by where   =          ,    =  2   /    , and   0 =    V  .It is clear that   is not symmetric and Equation ( 22) yields The second fundamental V-tensor   is defined as: The relative ℎand V-covariant derivatives of    and   are given by Let   (, ) be a vector field of   .The relative ℎand Vcovariant derivatives of   are given by Matsumoto [14] defined different kinds of hyperplanes and obtained their characteristic conditions, which are given in the following lemmas.
Theorem 5.The second fundamental v-tensor of Finsler hypersurface  −1 () 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 Applying (27) for the vector   , we get Using this and   | =     , (41) becomes Since   |  = − ℎ  ℎ  , using (33) and (40), we get Thus (43) gives Since   is symmetric, it is clear that  | is symmetric.Further contracting (45) with V  and then with V  , we get √  2 1 + 2

Lemma 1 .
A hypersurface  −1 is a hyperplane of the first kind if and only if   = 0 or equivalently  0 = 0. Lemma 2. A hypersurface  −1 is a hyperplane of the second kind if and only if   = 0. Lemma 3. A hypersurface  −1 is a hyperplane of the third kind if and only if   = 0 =   .