Integrable geodesic flows on orientable two-dimensional surfaces and topological billiards

V. V. Vedyushkina (Fokicheva) ; A. T. Fomenko

Izvestiya. Mathematics , Tome 83 (2019) no. 6, pp. 1137-1173.

Résumé

The authors have recently introduced the class of topological billiards. Topological billiards are glued from elementary planar billiard sheets (bounded by arcs of confocal quadrics) along intervals of their boundaries. It turns out that the integrability of the elementary billiards implies that of the topological billiards. We show that all classical linearly and quadratically integrable geodesic flows on tori and spheres are Liouville equivalent to appropriate topological billiards. Moreover, the linear and quadratic integrals of the geodesic flows reduce to a single canonical linear integral and a single canonical quadratic integral on the billiard. These results are obtained within the framework of the Fomenko–Zieschang theory of the classification of integrable systems.

Détail

Keywords: integrable system, topological billiard, geodesic flow
Mots-clés : Liouville equivalence, Fomenko–Zieschang invariant.

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