The canonical contact structure on the space of oriented null geodesics

YAMILE GODOY; MARCOS SALVAI

Rosario

Encuentro; Encuentro de Geometría Diferencial Rosario 2012; 2012

Facultad de Ciencias Exactas, Ingeniería y Agrimensura

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 $S$ be the pseudosphere of signature $(k,m)$. We show that $\mathcal{L}^{0}(S)$ 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 $\mathcal{L}^{0}(N)$ to be a manifold, and exhibit a contactomorphism with a concrete contact manifold.