  Many classical facts in Riemannian geometry have their pseudo-Riemannian
analogs. For instance, the spaces of space-like and time-like geodesics on a
pseudo-Riemannian manifold have natural symplectic structures (just like in the
Riemannian case), while the space of light-like geodesics has a natural contact
structure. We discuss the geometry of these structures in detail, as well as
introduce and study pseudo-Euclidean billiards. In particular, we prove
pseudo-Euclidean analogs of the Jacobi-Chasles theorems and show the
integrability of the billiard in the ellipsoid and the geodesic flow on the
ellipsoid in a pseudo-Euclidean space.
