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Ellipsoids, complete integrability and hyperbolic geometry. (English) Zbl 1013.37029
This paper concerns two closely related dynamical systems; the billiard transformation inside the ellipsoid and the geodesic flow on the ellipsoid in Euclidean space. Both are classical examples of completely integrable systems. The author describes a new proof of the complete integrability of these systems given in Geom. Funct. Anal. 7, 549-608 (1993; Zbl 0970.37042) and Comment. Math. Helv. 74, 306-321 (1999; Zbl 0958.37008). An interesting feature of this proof is an unexpected connection with hyperbolic geometry and the same proof applies to the ellipsoid in a space of constant curvature, positive or negative. It is a byproduct of a new class of dynamical systems, discovered by the author, called projective billiards [Ergod. Theory Dyn. Syst. 17, 957-976 (1997; Zbl 0886.58064)] and is based on the construction of a metric on the ellipsoid whose unparametrized 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).
The paper is expository, and the proofs, especially the computational ones, are omitted.

37D40 Dynamical systems of geometric origin and hyperbolicity (geodesic and horocycle flows, etc.)
53D25 Geodesic flows in symplectic geometry and contact geometry
37J05 Relations of dynamical systems with symplectic geometry and topology (MSC2010)
53A20 Projective differential geometry