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

YAMILE GODOY; 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 $mathcal{L}^{0}(N)$, the space of all its oriented null geodesics, is a manifold. B. Khesin and S. Tabachnikov introduce a canonical contact structure on $mathcal{L}^{0}(N)$ (generalizing the definition 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 $mathcal{L}^{0}(S^{k,m})$ 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 $mathcal{L}^{0}(S^{k,m})$ associated with some simple regions in $S^{k,m}$ in terms of the geodesic flows of spheres. For $N$ the pseudo-Riemannian product of two complete Riemannian manifolds, we give geometrical conditions on the factors for $mathcal{L}^{0}(N)$ to be a manifold and exhibit a contactomorphism with some standard contact manifold.