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

Let S k,m be the pseudosphere of signature ( k,m ). We show that the space ℒ 0 (S k,m ) of all oriented null geodesics in S k,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 Sk;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,m ) associated with some simple regions in Sk;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.