The canonical contact structure on the space of oriented null geodesics

YAMILE GODOY ; MARCOS SALVAI

San Pablo

Seminario; Seminario del grupo de Geometría Diferencial del IME-USP; 2012

Instituto de Matemática e Estatística (IME) - USP

Let N be a pseudo-Riemannian manifold such that Lº(N), the space of all its oriented null geodesics, is a manifold. B. Khesin and S. Tabachnikov introduce a canonical contact structure on Lº(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 be the pseudosphere of signature (k,m). We show that Lº(S) is a manifold and find a contactomorphism with some standard contact manifold, namely, the unit tangent bundle of some pseudo-Riemannian manifold. We present an application to the null billiard operator. For N the pseudo-Riemannian product of two Riemannian manifolds, we give geometrical conditions on the factors for Lº(N) to be a manifold, and exhibit a contactomorphism with a concrete contact manifold.