On the definition and examples of Finsler metrics

Abstract

For a standard Finsler metric F on a manifold M, the domain is the whole tangent bundle T M and the fundamental tensor g is positive-definite. However, in many cases (for example, for the well-known Kropina and Matsumoto metrics), these two conditions hold in a relaxed form only, namely one has either a pseudo-Finsler metric (with arbitrary g) or a conic Finsler metric (with domain a “conic” open domain of T M). Our aim is twofold. First, we want to give an account of quite a few subtleties that appear under such generalizations, say, for conic pseudo-Finsler metrics (including, as a preliminary step, the case of Minkowski conic pseudo-norms on an affine space). Second, we aim to provide some criteria that 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, more precisely, that the conic domain on which g remains positive-definite. Such a combination generalizes the known (↵, )-metrics in different directions. Remarkably, classical examples of Finsler metrics are reobtained and extended, with explicit computations of their fundamental tensors.