Certain generalizations of Finsler metrics

Abstract

Scrutinizing the main body of results from Finsler geometry it is observed that many of them depend on the Finsler metric only and not on the fundamental Finsler function. Moreover, there are many such results in which only some basic properties of the Finsler metric are involved. These facts led R. Miron ([9] [11]) to propose the study of generalized Lagrange metrics, GL–metrics for brevity, whose definition is tailored after the basic properties of Finsler metrics. The geometry of these metrics proved to be useful in the Theory of General Relativity, Gauge Theory, and Ecology (cf. [11] and references therein). Certain problems from Mechanics and Theoretical Physics require one, even at the Finslerian level, to study the geometry of a GL–metric which, furthermore, depends on a special variable analogous to the physical time. We contributed to this study in [4]. The main objective of this paper is to review from our own viewpoint the generalizations of Finsler metrics mentioned above. We take this opportunity to cast a new light on some well–known results and to add several new ones. Some new examples are provided, too.