"A detailed study of the notions of convexity for a hypersurface in a Finsler manifold is carried out. In particular, the infinitesimal and local notions of convexity are shown to be equivalent. Our approach differs from Bishop's one in his classical result (Bishop, Indiana Univ Math J 24:169-172, 1974) for the Riemannian case. Ours not only can be extended to the Finsler setting but it also reduces the typical requirements of differentiability for the metric and it yields consequences on the multiplicity of connecting geodesics in the convex domain defined by the hypersurface."

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 x S and Randers metrics on S. In particular: (1) For stationary spacetimes: we give a simple characterization of when R x 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 metric R 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.

Given a Riemannian manifold (M, g) and an embedded hypersurface H in M, a result by R. L. Bishop states that infinitesimal convexity on a neighborhood of a point in H implies local convexity. Such result was extended very recently to Finsler manifolds by the author et al. in [2]. We show in this note that the techniques in [2], unlike the ones in Bishop’s paper, can be used to prove the same result when (M, g) is semi-Riemannian. We make some remarks for the case when only time-like, null, or space-like geodesics are involved. The notion of geometric convexity is also reviewed, and some applications to geodesic connectedness of an open subset of a Lorentzian manifold are given.

We give the details of the proof of the existence of an isomorphism between some homological groups related to the critical groups of the energy functional of a Finsler manifold in the two end-points boundary conditions.

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 existence and multiplicity results for light rays, using the Finsler theory. Finally the case of timelike geodesics with fixed energy is considered. The research that led to the present paper was partially supported by a grant of the group GNAMPA of INdAM