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

M. K. Gupta,¹ Abhay Singh,¹ and P. N. Pandey²

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 [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 -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.

References

M. Matsumoto, “On C-reducible finsler spaces,” Tensor N.S., vol. 24, pp. 29–37, 1972.

M. Hashiguchi, S.-i. Hōjō, and M. Matsumoto, “On Landsberg spaces of two dimensions with -metric,” Journal of the Korean Mathematical Society, vol. 10, pp. 17–26, 1973.

M. Matsumoto, “Projectively flat Finsler spaces with -metric,” Reports on Mathematical Physics, vol. 30, no. 1, pp. 15–20, 1991.

C. Shibata, “On Finsler spaces with an -metric,” Journal of Hokkaido University of Education. Section II A, vol. 35, no. 1, pp. 1–16, 1984.

P. L. Antonelli, A. Bóna, and M. A. Slawiński, “Seismic rays as Finsler geodesics,” Nonlinear Analysis RWA, vol. 4, no. 5, pp. 711–722, 2003.

P. L. Antonelli, R. S. Ingarden, and M. Matsumoto, The Theory of Sprays and Finsler Spaces with Applications in Physics and Biology, Kluwer Academic, Dordrecht, The Netherlands, 1993.

M. Rafie-Rad and B. Rezaei, “On Einstein Matsumoto metrics,” Nonlinear Analysis RWA, vol. 13, no. 2, pp. 882–886, 2012.

H. Shimada and S. V. Sabau, “Introduction to Matsumoto metric,” Nonlinear Analysis, vol. 63, pp. 165–168, 2005.

M. Matsumoto, “A slope of a mountain is a Finsler surface with respect to a time measure,” Journal of Mathematics of Kyoto University, vol. 29, no. 1, pp. 17–25, 1989.

A. Tayebi, E. Peyghan, and H. Sadeghi, “On Matsumoto-type Finsler metrics,” Nonlinear Analysis, vol. 13, no. 6, pp. 2556–2561, 2012.

M. Hashiguchi and Y. Ichijyo, “Randers spaces with rectilinear geodesics,” Reports of the Faculty of Science of Kagoshima University, no. 13, pp. 33–40, 1980.

C. Shibata, “On invariant tensors of -changes of Finsler metrics,” Journal of Mathematics of Kyoto University, vol. 24, no. 1, pp. 163–188, 1984.

H. G. Nagaraja and P. Kumar, “On Randers change of Matsumoto metric,” Bulletin of Mathematical Analysis and Applications, vol. 4, no. 1, pp. 148–155, 2012.

M. Matsumoto, “The induced and intrinsic Finsler connections of a hypersurface and Finslerien projective geometry,” Journal of Mathematics of Kyoto University, vol. 25, no. 1, pp. 107–144, 1985.

M. K. Gupta and P. N. Pandey, “On hypersurface of a Finsler space with a special metric,” Acta Mathematica Hungarica, vol. 120, no. 1-2, pp. 165–177, 2008.

M. K. Gupta and P. N. Pandey, “Hypersurfaces of conformally and -conformally related Finsler spaces,” Acta Mathematica Hungarica, vol. 123, no. 3, pp. 257–264, 2009.

U. P. Singh and B. Kumari, “On a hypersurface of a Matsumoto space,” Indian Journal of Pure and Applied Mathematics, vol. 32, no. 4, pp. 521–531, 2001.