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

YAMILE GODOY; MARCOS SALVAI

Córdoba

Congreso; IV Congreso Latinoamericano de Matemáticos (CLAM); 2012

Unión Matemática de América Latina y el Caribe; FaMAF-UNC; CIEM

ABSTRACT: 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.