MOSCOW MATHEMATICAL JOURNAL Volume 2, Number 1, January–March 2002, Pages 185–198 ELLIPSOIDS, COMPLETE INTEGRABILITY AND HYPERBOLIC GEOMETRY SERGE TABACHNIKOV Dedicated to the memory of J.Moser Abstract. We describe a new proof of the complete integrability of the two related dynamical systems: the billiard inside the ellipsoid and the geodesic flow on the ellipsoid (in Euclidean, spherical or hyperbolic space). The proof is based on the construction of a metric on the el- lipsoid whose nonparameterized geodesics coincide with those of the standard metric. This new metric is induced by the hyperbolic metric inside the ellipsoid (the Caley–Klein model of hyperbolic space). 2000 Math. Subj. Class. 53A15, 53A20, 53D25. Key words and phrases. Riemannian and Finsler metrics, symplectic and contact structures, geodesic flow, mathematical billiard, hyperbolic metric, Caley–Klein model, exact transverse line fields. 1. Introduction This paper concerns two closely related dynamical systems: the billiard trans- formation inside the ellipsoid and the geodesic flow on the ellipsoid in Euclidean space; both provide classical examples of completely integrable systems. We de- scribe a new proof of the complete integrability of these systems given in [27, 31]; an interesting feature of this proof is a rather unexpected connection with hyperbolic geometry. The same proof applies to the ellipsoid in a space of constant curvature, positive or negative. The proof is a byproduct of a study of a new class of dynami- cal systems, called projective billiards, and related problems of symplectic, Finsler and projective geometry, see [28, 29, 30, 32, 33]; we will discuss relevant part of this theory below. The geodesic flow on a Riemannian manifold M n is a flow on the tangent bundle TM : a tangent vector v moves with constant speed along the geodesic tangent to v. From the physical viewpoint, the geodesic flow describes the motion of a free particle on M . Identifying the tangent and cotangent bundles by the metric, the geodesic flow becomes a Hamiltonian vector field on the cotangent bundle T * M with its canonical symplectic structure “dp ∧ dq”, the Hamiltonian function being Received October 30, 2001; in revised form January 15, 2002. Partially supported by NSF grant DMS-9802849 and BSF grant. c 2002 Independent University of Moscow 185