Closed geodesics on compact Lorentzian manifolds of splitting type

In this paper, we consider the problem of the existence of a spacelike closed geodesic on compact Lorentzian manifolds. Tipler and Galloway proved that, under suitable topological properties of the manifold, there exists a closed timelike geodesic. In their proofs, they use the hypothesis that the time coordinate of one timelike geodesic has derivative always different from zero. This clearly fails for spacelike geodesics. Using variational methods and applying the relative category theory, we prove the existence of a closed spacelike geodesic on a compact manifold script M sign of splitting type. Observe that, thanks to the previous results, the existence of at least two geometrically distinct closed geodesics on script M sign follows.