On an R-Randers th-Root Space

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 [6–8]. 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 Nimine？ [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.