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Geometry
Volume 2013 (2013), Article ID 842573, 6 pages
http://dx.doi.org/10.1155/2013/842573
Research Article

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

1Department of Pure & Applied Mathematics, Guru Ghasidas Vishwavidyalaya, Bilaspur, India
2Department of Mathematics, University of Allahabad, Allahabad, India

Received 29 January 2013; Accepted 11 June 2013

Academic Editor: Anna Fino

Copyright © 2013 M. K. Gupta et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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 -metric was introduced by Matsumoto [1]. It was later discussed by several authors such as Shibata and others [24]. It has plenty of applications in various fields such as physics, mechanics, seismology, biology, and ecology [58]. Matsumoto introduced a special type of -metric of the form , , and , 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 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 . 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 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 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 is given by where The fundamental metric tensor is given by where Moreover, the reciprocal tensor of is given by where The Cartan tensor 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 , 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 , , and .

3. Induced Cartan Connection

A hypersurface of the underlying manifold may be represented parametrically by , where are the Gaussian coordinates on (Latin indices run from 1 to , while Greek indices take values from 1 to ). We assume that the matrix of projection factors is of rank . If the supporting element at a point of is assumed to be tangent to , we may then write so that is thought of as the supporting element of at the point . Since the function gives rise to a Finsler metric on , we get an ()-dimensional Finsler space . The metric tensor and the Cartan tensor are given by At each point of , a unit normal vector is defined by For the angular metric tensor , we have The inverse projection factors of are defined as where is the inverse of the metric tensor of .

From (17) and (19), it follows that and further For the induced Cartan connection on , the second fundamental -tensor and the normal curvature vector are given by where , , and . It is clear that is not symmetric and Equation (22) yields The second fundamental -tensor is defined as: The relative - and -covariant derivatives of and are given by Let be a vector field of . The relative - and -covariant derivatives of are given by

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

Lemma 1. A hypersurface is a hyperplane of the first kind if and only if or equivalently .

Lemma 2. A hypersurface is a hyperplane of the second kind if and only if .

Lemma 3. A hypersurface is a hyperplane of the third kind if and only if .

4. Hypersurface of the Finsler Space with Randers Change of Matsumoto Metric

Let us consider the Randers change of Matsumoto metric with the gradient for a scalar function and a hypersurface given by the equation (constant). From parametric equation of , we get , so that are regarded as covariant components of a normal vector field of . Therefore, along , we have Therefore the induced metric of is given by which is a Riemannian metric.

At a point of , from (6), (8), and (10), we have Therefore (9) gives using (28) we get which gives where is the length of the vector . Using (31) and (33) we get

Theorem 4. Let be a Finsler space with Randers change of Matsumoto metric with a gradient and let be a hypersurface of , which is given by (constant). Then the induced metric on is Riemannian metric given by (29), and the scalar function is given by (33) and (34).

Along , the angular metric tensor and the metric tensor of are given by If denote the angular metric tensor of the Riemannian metric , then, using (28), (35), and (18), we get From (8), we get Thus, along , , and therefore (12) gives . Then the Cartan tensor becomes and therefore, using (18), (25), and (28), we get and hence from (23) it follows that is symmetric. Thus we have the following.

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

Taking -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 , using (33) and (40), we get Thus (43) gives Since is symmetric, it is clear that is symmetric. Further contracting (45) with and then with , we get In view of Lemma 1, the hypersurface is a hyperplane of the first kind if and only if . Here being the covariant derivative with respect to the Cartan connection of may depend on .

Since is a gradient vector, from (13), we have . Thus (14) reduces to In view of (30) and (31), the relations in (15) become to By virtue of (28) we have , which leads . Therefore we have Using the relation (28), we get is the covariant derivative of with respect to relative to the Cartan connection of , and is the covariant derivative of with respect to relative to the Riemannian connection: Using (51), we get Consequently, (46) may be written as Thus the condition is equivalent to , where does not depend upon . Since is to satisfy (28), the condition is written as for some , so that we have Using (28), it follows that Again (48) and (55) gives Using the (25), (31), (34), (40), and (47), we get Therefore in view of (52), (45) reduces to

Theorem 6. The necessary and sufficient condition for the hypersurface 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 -tensor of hypersurface is proportional to its angular metric tensor.

In view of Lemma 2, the hypersurface is a hyperplane of the second kind if and only if . Thus from (59) we get . Therefore there exists a function such that . Thus (55) gives

Theorem 7. The necessary and sufficient condition for the hypersurface 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 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.

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