The canonical contact structure on the space of oriented null geodesics of pseudo-spheres and products

GODOY, YAMILE; MARCOS SALVAI

ADVANCES IN GEOMETRY

WALTER DE GRUYTER & CO

Lugar: Berlin; Año: 2013 vol. 13 p. 713 - 722

1615-715X

Let N be a pseudo-Riemannian manifold such that L_0(N), the space of all its oriented null geodesics, is a manifold. B. Khesin and S. Tabachnikov introduce a canonical contact structure on L_0(N) (generalizing the denition given by R. Low in the Lorentz case), and study it for the pseudo-Euclidean space. We continue in that direction for other spaces. Let S^(k,m) be the pseudosphere of signature (k,m). We show that L_0(S^(k,m)) is a manifold and describe geometrically its canonical contact distribution in terms of the space of oriented geodesics of certain totally geodesic degenerate hypersurfaces in S^(k,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 S^(k,m) in terms of the geodesic ows of spheres. For N the pseudo-Riemannian product of two complete Riemannian manifolds, we give geometrical conditions on the factors for L_0(N) to be a manifold and exhibit a contactomorphism with some standard contact manifold.