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.

We introduce the notion of a standard static Finsler spacetime where the base is a Finsler manifold. We prove some results which connect causality with the Finslerian geometry of the base extending analogous ones for static and stationary Lorentzian spacetimes.Comment: AMS-LaTeX, 22 pages, 1 figure. v2 matches the published versio

The notion of wind Finslerian structure Σ is developed; this is a generalization of Finsler metrics (and Kropina ones) where the indicatrices at the tangent spaces may not contain the zero vector. In the particular case that these indicatrices are ellipsoids, called here wind Riemannian structures, they admit a double interpretation which provides: (a) a model for classical Zermelo's navigation problem even when the trajectories of the moving objects (planes, ships) are influenced by strong winds or streams, and (b) a natural description of the causal structure of relativistic spacetimes endowed with a non-vanishing Killing vector field K (SSTK splittings), in terms of Finslerian elements. These elements can be regarded as conformally invariant Killing initial data on a partial Cauchy hypersurface. The links between both interpretations as well as the possibility to improve the results on one of them using the other viewpoint are stressed.The wind Finslerian structure Σ is described in terms of two (conic, pseudo) Finsler metrics, F and F l , the former with a convex indicatrix and the latter with a concave one. Notions such as balls and geodesics are extended to Σ. Among the applications, we obtain the solution of Zermelo's navigation with arbitrary time-independent wind, metric-type properties for Σ (distance-type arrival function, completeness, existence of minimizing, maximizing or closed geodesics), as well as description of spacetime elements (Cauchy developments, black hole horizons) in terms of Finslerian elements in Killing initial data. A general Fermat's principle of independent interest for arbitrary spacetimes, as well as its applications to SSTK spacetimes and Zermelo's navigation, are also provided.

We show that the index of a lightlike geodesic in a conformally standard stationary spacetime (M 0 × R, g ) is equal to the index of its spatial projection as a geodesic of a Finsler metric F on M 0 associated to (M 0 × R, g ). Moreover we obtain the Morse relations of lightlike geodesics connecting a point p to a curve γ(s) = (q 0 , s) by using Morse theory on the Finsler manifold (M 0 ,F ). To this end, we prove a splitting lemma for the energy functional of a Finsler metric. Finally, we show that the reduction to Morse theory of a Finsler manifold can be done also for timelike geodesics.2000 Mathematics Subject Classification. 53C22, 53C50, 53C60, 58E05.