On the definition and examples of Finsler metrics
  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. 
Miguel Angel Javaloyes and Miguel S\'anchez
math.DG