Pseudo-Riemannian geodesics and billiards

Details Pseudo-Riemannian geodesics and billiards Pseudo-Riemannian geodesics and billiards

2006-08-24

arxiv.org/abs/math/0608620v3

Mathematics

Differential Geometry

title abbreviated, text edited; to appear in Advances in Mathematics

Scientific paper

Many classical facts in Riemannian geometry have their pseudo-Riemannian analogs. For instance, the spaces of space-like and time-like geodesics on a pseudo-Riemannian manifold have natural symplectic structures (just like in the Riemannian case), while the space of light-like geodesics has a natural contact structure. We discuss the geometry of these structures in detail, as well as introduce and study pseudo-Euclidean billiards. In particular, we prove pseudo-Euclidean analogs of the Jacobi-Chasles theorems and show the integrability of the billiard in the ellipsoid and the geodesic flow on the ellipsoid in a pseudo-Euclidean space.

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