Proceedings of the Royal Society of Edinburgh, 128A, 447-462, 1998 Closed geodesies on compact Lorentzian manifolds of splitting type Flavia Antonacci Dipartimento di Matematica, Universita degli Studi di Roma III, Largo S. Leonardo Murialdo 1, Italy e-mail: flavia@matrm3.mat.uniroma3.it Rosella Sampalmieri Facolta di Ingegneria, Dipartimento di Energetica, Universita dell'Aquila, Monteluco di Roio, L'Aquila, Italy e-mail: sampalm@ing.univaq.it (MS received 13 September 1996. Revised MS received 30 January 1997) 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 geodesies. Using variational methods and applying the relative category theory, we prove the existence of a closed spacelike geodesic on a compact manifold M of splitting type. Observe that, thanks to the previous results, the existence of at least two geometrically distinct closed geodesies on M follows. 1. Introduction and statement of the main results The problem of the existence of closed geodesies on compact Riemannian manifolds has been thoroughly investigated (see for instance [11]) and there are also some results in the noncompact case [2,20]. On a compact Lorentzian manifold, the situation is different. Indeed, as far as the authors know, the only answers for this problem are contained in [7,8,12,21]. In [7,21] it has been proved that under suitable topological properties of the manifold, there is a closed timelike geodesic. In [8] the author establishes the existence of a closed timelike or lightlike geodesic on every two-dimensional compact Lorentzian manifold and gives an example of a compact manifold without any closed spacelike geodesies (see Remark 1.3, below). In [12] the existence of a closed spacelike geodesic was proved in the stationary case, and the results of [7, 8,21] were obtained using the causal theory of Lorentzian manifolds. In this paper, using different methods, namely global variational methods and the relative category theory, we prove the existence of a closed spacelike geodesic on a compact Lorentzian manifold of splitting type. Thanks to the results of [21] and [7] we have that in such a situation there are at least two closed geodesies: one timelike and one spacelike.