This article is cited in 19 papers

Ellipsoids, complete integrability and hyperbolic geometry

S. L. Tabachnikov

Pennsylvania State University

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 ellipsoid 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).

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.

MSC: 53A15, 53A20, 53D25

Received: October 30, 2001; in revised form January 15, 2002

Language: English

DOI: 10.17323/1609-4514-2002-2-1-183-196

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