Integrable billiards and quadrics

Abstract

Billiards inside quadrics are considered as integrable dynamical systems with a rich geometric structure. The two-way interaction between the dynamics of billiards and the geometry of pencils of quadrics in an arbitrary dimension is considered. Several well-known classical and modern genus-1 results are generalized to arbitrary dimension and genus, such as: the Poncelet theorem, the Darboux theorem, the Weyr theorem, and the Griffiths-Harris space theorem. A synthetic approach to higher-genera addition theorems is presented. © 2010 RAS(DoM) and LMS.