Pseudo-Riemannian geodesics and billiards

Boris Khesin, Serge Tabachnikov

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57 Scopus citations


In pseudo-Riemannian geometry 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. Furthermore, the space of all geodesics has a structure of a Jacobi manifold. We describe the geometry of these structures and their generalizations. We also introduce and study pseudo-Euclidean billiards, emphasizing their distinction from Euclidean ones. We present a pseudo-Euclidean version of the Clairaut theorem on geodesics on surfaces of revolution. 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.

Original languageEnglish (US)
Pages (from-to)1364-1396
Number of pages33
JournalAdvances in Mathematics
Issue number4
StatePublished - Jul 10 2009

All Science Journal Classification (ASJC) codes

  • General Mathematics


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