On the definition and examples of Finsler metrics

TL;DR

This paper explores generalized Finsler metrics, including pseudo-Finsler and conic Finsler types, providing criteria for when combined metrics retain Finsler properties and extending classical examples with explicit tensor computations.

Contribution

It offers a detailed account of subtleties in generalized Finsler metrics and criteria for their positivity domains, extending known metrics like (alpha,beta)-metrics.

Findings

Criteria for positivity of generalized Finsler metrics

Explicit computations of fundamental tensors for extended metrics

Reinterpretation and extension of classical Finsler examples

Abstract

For a standard Finsler metric F on a manifold M, its domain is the whole tangent bundle TM and its fundamental tensor g is positive-definite. However, in many cases (for example, the well-known Kropina and Matsumoto metrics), these two conditions are relaxed, obtaining then either a pseudo-Finsler metric (with arbitrary g) or a conic Finsler metric (with domain a "conic" open domain of TM). Our aim is twofold. First, to give an account of quite a few subtleties which appear under such generalizations, say, for conic pseudo-Finsler metrics (including, as a previous step, the case of Minkowski conic pseudo-norms on an affine space). Second, to provide some criteria which determine when a pseudo-Finsler metric F obtained as a general homogeneous combination of Finsler metrics and one-forms is again a Finsler metric ---or, with more accuracy, the conic domain where g remains positive definite. Such a combination generalizes the known (alpha,beta)-metrics in different directions. Remarkably, classical examples of Finsler metrics are reobtained and extended, with explicit computations of their fundamental tensors.