Math Seminars

Sergei Tabachnikov
University of Arkansas

Ellipsoids, Integrability and Hyperbolic Geometry

I will discuss new proofs of the two related classical facts: the geodesic flow on the ellipsoid and the billiard ball map inside it are completely integrable. It is well known that the geodesic flow and the billiard ball map preserve natural symplectic structures associated with the Euclidean geometry of the ambient space. The proofs are based on the observation that they also preserve symplectic structures associated with the projective model of the hyperbolic geometry inside the ellipsoid.