Differential geometry of generalized lagrangian functions

Abstract

There are many generalizations of Finsler geometry. A Finsler metric function is defined on the tangent bundle of a differentialble manifold with some assumptions. Especially, it is assumed to be positively homogeneous. The importance of a generalized metric has been emphasized by many authors ([2], [5], [7]). Some of them studied the non-homogeneous “metric” space ([1], [3], [4]). In [1], they investigated generalized Lagrangian space $(M, L)$ from the view point of Finsler spaces $(M^{*}, L^{*})$, where $M^{*}$ is the $(n+1)$-dimensional manifold and $L^{*}$ is positively homogeneous. The purpose of the present paper is to investigate the function without the assumption of homogeneity from another point of view.