On the definition and examples of Finsler metrics

• Miguel Angel Javaloyes ^[1] ; Miguel Sánchez ^[2]
  1. [1] Universidad de Murcia

     Universidad de Murcia

     Murcia, España

     
  2. [2] Universidad de Granada

     Universidad de Granada

     Granada, España

     
• Localización: Annali della Scuola Normale Superiore di Pisa. Classe di scienze, ISSN 0391-173X, Vol. 13, Nº 3, 2014, págs. 813-858
• Idioma: inglés
• DOI: 10.2422/2036-2145.201203_002
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• Resumen

  • 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.