The canonical contact structure on the space of oriented null geodesics of pseudospheres and products

Let S-k,S-m be the pseudosphere of signature (k, m). We show that the space L-0 (S-k,S-m) of all oriented null geodesics in S-k,S-m is a manifold, and we describe geometrically its canonical contact distribution in terms of the space of oriented geodesics of certain totally geodesic degenerate hypersurfaces in S-k,S-m. Further, we find a contactomorphism with some standard contact manifold, namely, the unit tangent bundle of some pseudo-Riemannian manifold. Also, we express the null billiard operator on L-0 (S-k,S-m) associated with some simple regions in S-k,S-m in terms of the geodesic flows of spheres. For the pseudo-Riemannian product N of two complete Riemannian manifolds, we give geometrical conditions on the factors for L-0 (N) to be manifolds and exhibit a contactomorphism with some standard contact manifold.