We obtain some results in both Lorentz and Finsler geometries, by using a correspondence between the conformal structure (Causality) of standard stationary spacetimes on M = R × S and Randers metrics on S. In particular:(1) For stationary spacetimes: we give a simple characterization of when R × S is causally continuous or globally hyperbolic (including in the latter case, when S is a Cauchy hypersurface), in terms of an associated Randers metric. Consequences for the computability of Cauchy developments are also derived.(2) For Finsler geometry: Causality suggests that the role of completeness in many results of Riemannian Geometry (geodesic connectedness by minimizing geodesics, Bonnet-Myers, Synge theorems) is played by the compactness of symmetrized closed balls in Finslerian Geometry. Moreover, under this condition we show that for any Randers metric R there exists another Randers metricR with the same pregeodesics and geodesically complete.Even more, results on the differentiability of Cauchy horizons in spacetimes yield consequences for the differentiability of the Randers distance to a subset, and vice versa.2000 Mathematics Subject Classification. 53C22, 53C50, 53C60, 58B20.

In this paper we first study some global properties of the energy functional on a non-reversible Finsler manifold. In particular we present a fully detailed proof of the Palais-Smale condition under the completeness of the Finsler metric. Moreover, we define a Finsler metric of Randers type, which we call Fermat metric, associated to a conformally standard stationary spacetime. We shall study the influence of the Fermat metric on the causal properties of the spacetime, mainly the global hyperbolicity. Moreover, we study the relations between the energy functional of the Fermat metric and the Fermat principle for the light rays in the spacetime. This allows one to obtain E. Caponio and A. Masiello are supported by M.I.U.R. Research project PRIN07 "Metodi Variazionali e Topologici nello Studio di Fenomeni Nonlineari". M. Á.

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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.