Some results about geodesics on Lorentzian manifolds of splitting type

Let us consider a Lorentzian manifold (M,g), that is a smooth, connected, finite-dimensional manifold M endowed with a smooth symmetric second order metric tensor field g, having index 1. For any z in M, it induces on T,M a nondegenerate bilinear form g(z)[., .] having exactly one negative eigenvalue. In order to study the geometry of (M,g) geodesic curves play an important role. We recall that a geodesic is a smooth curve 7: [0, l] + M which satisfies the equation: v,j-0, where V,+(s) denotes the covariant derivative of 7(s) with respect to the Levi-Civita connection of the metric tensor g. A closed geodesic is a geodesic 7 such that r(O) = 7(l) and %O) = ?(I). It is well known that if 7 is a geodesic, there exists a constant E-, such that E-, = g(7(s))[j(s),j(s)] for all s E [0, l] . Then 7 is said timelike, lightlike or spacelike respectively when E, is negative, null or positive. The aim of this paper is to review some recent results concerning some properties of geodesics on a Lorentzian manifold. Particularly we will analyse two questions: the existence of closed geodesics and the geodesical connectedness for some classes of Lorentzian manifolds. We recall (M, g) is said geodesicallyconnectedif every couple of its points can be joined by a geodesic. The results here reported are obtained by means of global variational methods and relative category theory and they refer to [l], [2], [4], [5], [6], [12], [13], [15]. The paper will be divided into two sections, dealing separately with the previous problems.