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Geometry
Volume 2013 (2013), Article ID 649168, 7 pages
http://dx.doi.org/10.1155/2013/649168
Research Article

On an R-Randers th-Root Space

Department of Mathematics, University of Allahabad, Allahabad 211002, India

Received 21 October 2012; Revised 14 January 2013; Accepted 14 January 2013

Academic Editor: Matthew He

Copyright © 2013 P. N. Pandey and Shivalika Saxena. 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

We consider an n-dimensional Finsler space with the metric , where is an th-root metric and is a Riemannian metric. We call such space as an R-Randers th-root space. We obtain the expressions for the fundamental metric tensor, Cartan tensor, geodesic spray coefficients, and the coefficients of nonlinear connection in an R-Randers th-root space. Some other properties of such space have also been discussed.

1. Introduction

The theory of an mth-root metric was introduced by Shimada [1] in 1979. By introducing the regularity of the metric, various fundamental quantities of a Finsler metric could be found. In particular, the Cartan connection of a Finsler space with mth-root metric was introduced from the theoretical standpoint. Matsumoto and Okubo [2] studied Berwald connection of a Finsler space with mth-root metric and gave main scalars in two dimensional cases and defined higher-order Christoffel symbols. The mth-root metric is used in many problems of theoretical physics [3]. Pandey et al. [4] studied three-dimensional Finsler space with mth-root metric. To discuss general relativity with the electromagnetic field, Randers [5] introduced a metric of the form , where is a square root metric and is a differential one form. In his honor, this metric is called Randers metric, and it has been extensively studied by several geometers and physicists [68].

Munteanu and Purcaru [9] first defined complex Finsler spaces by reducing the scalar in the homogeneity condition of fundamental function, that is, , and named such space as an R-complex Finsler space. Aldea and Purcaru [10] introduced the concept of R-complex Finsler spaces with -metrics. In 2011, Purcaru [11] also discussed the notion of R-complex Finsler space with Kropina metric. Lungu and Nimineţ [12] studied a special Finsler space with the metric of the form , where is a quartic root metric and is a square root metric. They regarded this space as an R-Randers quartic space and obtained many results related to it. The aim of the present paper is to study a more general space with the metric , where is an mth-root metric and is a Riemannian metric. We call the space endowed with this metric as an R-Randers mth-root space.

The paper is organized as follows. Section 2 deals with some preliminary concepts required for the discussion of the following sections. It includes the notion of an R-Randers mth-root space. In Section 3, we derive certain identities satisfied in an R-Randers mth-root space. We obtain the fundamental metric tensor , its inverse , and the Cartan tensor for an R-Randers mth-root space. In Section 4, we obtain the spray coefficients of an R-Randers mth-root space. It includes the equations of the geodesics. Section 5 discusses the nonlinear connection in an R-Randers mth-root space.

2. Preliminaries

Let be an n-dimensional Finsler space. The mth-root metric on is defined as , where are components of an mth-order covariant symmetric tensor.

In case of , the metric is Riemannian, and in the cases and these metrics are called cubic and quartic, respectively.

The covariant symmetric metric tensor of is defined by This tensor is positively homogeneous of degree zero in . From the metric tensor , we construct the Cartan tensor by The tensor is symmetric in its lower indices and is positively homogeneous of degree in . Due to its homogeneous and symmetric properties, it satisfies the following: The angular metric tensor of a Finsler space is written as where .

The geodesic of a Finsler space is given by where is the geodesic spray coefficients given by The nonlinear connection of a Finsler space is defined as In the present paper, we study the space whose fundamental function is given by where is  an  mth-root  metric and is a Riemannian metric. We call this space an R-Randers mth-root space.

3. Fundamental Metric Tensor and Cartan Tensor

In this section, we find the fundamental metric tensor , its inverse , angular metric tensor , and the Cartan tensor for an R-Randers mth-root space.

Differentiating (9) partially with respect to , we get where .

Differentiating (11) partially with respect to , we find where .

Take and , where the subscripts denote the degree of homogeneity of the corresponding entities with respect to . In view of this, (11) and (12) give where Now, differentiating (10) partially with respect to , we get where and .

Further differentiating (16) partially with respect to , we find where .

Differentiating partially with respect to , we get Thus, we have the following.

Proposition 1. In an R-Randers mth-root space, the following identities hold:

Differentiating (8) partially with respect to and using (13) and (16), we have Further, differentiating (20) partially with respect to and using (14) and (17), we get From (1), we have Using (20) and (21), (22) yields Let us take and , where and satisfy The matrix is nonsingular, in general, for 's are functions of and whereas 's are functions of only.

Equation (23) takes the form Thus, we have the following.

Theorem 2. The fundamental metric tensor of an R-Randers mth-root space is given by (25).

Theorem 3. In an R-Randers mth-root space, the inverse of the fundamental metric tensor is given by where and .

Proof. Let be the inverse of nonsingular matrix . Suppose that is given by (26).
Now,
Therefore, given by (26) is the inverse of the matrix . This also shows that is nondegenerate.

Using (20) and (23) in (4), we get the angular metric tensor of an R-Randers mth-root space: Thus, we have the following.

Theorem 4. In an R-Randers mth-root space, the angular metric tensor is given by (28).

Differentiating partially with respect to , we get where .

Also, we have where .

Thus, we have the following.

Proposition 5. In an R-Randers mth-root space, the following holds good:

Differentiating (23) partially with respect to , we get

Partial differentiation of and , with respect to , yields Further, differentiation of and with respect to gives where .

Thus, we have the following.

Proposition 6. In an R-Randers mth-root space the following holds good:

If we use (2), (15), (18), (20), (29), (30), (33), and (34) in (32), on simplification it follows that where Thus, we have the following.

Theorem 7. In an R-Randers mth-root space, the Cartan tensor is given by (36).

4. Spray and Equation of Geodesics

In this section, we discuss the spray of an R-Randers mth-root space and obtain its local coefficients. We also obtain the equation of geodesics in such space.

If we differentiate (9) partially with respect to , we get where Further, differentiating (13) partially with respect to and utilizing , we have Differentiating (10) partially with respect to , we get where Also, we have Next, differentiating (16) partially with respect to and using (43), we obtain Thus, we have the following.

Proposition 8. In an R-Randers mth-root space, the following holds good:

If we differentiate (8) partially with respect to and use (38) and (41), it follows that Next, differentiating (20) partially with respect to and using (40) and (44), we get In view of (46) and (47), (6) gives that is where and is given by (26). Thus, we have the following.

Theorem 9. In an R-Randers mth-root space, the spray coefficients are given by (49).

In view of (5) and Theorem 9, we have the following.

Corollary 10. In an R-Randers mth-root space, the equation of geodesics is given by where the spray coefficients are given by (49).

5. Nonlinear Connection

In this section, we obtain the coefficients of nonlinear connection of the space under consideration.

Differentiating (49) partially with respect to , we have Differentiation of partially with respect to yields where .

Next, differentiating partially with respect to and using (15), (20), and (29), we get where .

Differentiating and partially with respect to , we, respectively, have Also, we have Using (7), (15), (18), (29), (30), (33), and (53)–(56) in (52), we have

Thus, we have the following.

Theorem 11. The local coefficients of the nonlinear connection of an R-Randers mth-root space are given by (57).

Acknowledgments

The authors are thankful to the reviewers for their valuable comments and suggestions. S. Saxena gratefully acknowledges the financial support provided by the UGC, Government of India.

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