Lagrange Spaces with -Metric

We study Lagrange spaces with -metric, where is a cubic metric and is a 1-form. We obtain fundamental metric tensor, its inverse, Euler-Lagrange equations, semispray coefficients, and canonical nonlinear connection for a Lagrange space endowed with a -metric. Several other properties of such space are also discussed. 1. Introduction Finsler spaces endowed with -metric were studied by several geometers such as Matsumoto [1, 2] and Kitayama et al. [3], and various important applications of such spaces have been observed in physics and relativity theory (cf. [4, 5]). The notion of -metric was taken to a more general space called Lagrange space and the study was performed by the authors such as Miron [6], Nicolaescu [7, 8], and the present authors [9]. An -dimensional Lagrange space is said to be endowed with -metric if Lagrangian is function of and only, that is, being a Riemannian metric and a 1-form. Recently, Pandey and Chaubey [10] discussed Lagrange spaces with -metric and obtained several results. They called a Lagrange space to be endowed with -metric if Lagrangian is a function of and only, that is, where is a cubic metric and is a 1-form, that is, and . The paper [10] by Pandey and Chaubey is full of flaws and needs to be rectified. The aim of the present paper is to develop a revised and modified theory of Lagrange spaces with -metric. The paper is organized as follows. In Section 2, we define a Lagrange space and discuss some preliminary results required for the discussion of the following sections. It includes the notion of a Lagrange space with -metric. In Section 3, we discuss some properties of a Lagrange space with -metric and obtain the expression for the fundamental metric tensor and its inverse . In Section 4, we consider the variational problem in Lagrange spaces with -metric and obtain various forms of Euler-Lagrange equations. Section 5 deals with the semispray of a Lagrange space with -metric. In Section 6, we obtain the coefficients of nonlinear connection in a Lagrange space endowed with -metric. Section 7 consists of concluding remarks on the results obtained in the paper. 2. Preliminaries Let be an -dimensional smooth manifold and let be its tangent bundle. Let and be local coordinates on and , respectively. A Lagrangian is a function which is a smooth function on and continuous on the null section. The Lagrangian is said to be regular if rank？？ , where is a covariant symmetric tensor called the fundamental metric tensor of the Lagrangian . A Lagrange space is a pair being a regular Lagrangian whose metric tensor has constant