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

Abstract

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.

Billiards and integrable systems

“…A long list of publications on this subject was presented in books by Fomenko [78] and Bolsinov and Fomenko [6], and in papers by Fomenko and Vedyushkina [55], [79], and Fomenko and Kibkalo [34], [66].…”

“…Recently Vedyushkina and Fomenko showed [55] that, given a closed orientable surface and an arbitrary geodesic flow on it which is linearly or quadratically integrable, we can algorithmically construct a billiard (a topological billiard or a billiard book) such that the geodesic flow and the billiard have the same Fomenko-Zieschang invariants.…”

“…Assume that a geodesic flow on a torus has a first integral that is quadratic in momenta and the metric on the torus is globally Liouvillean. Using the result of Fomenko and Vedyushkina in [55] we construct algorithmically a confocal billiard book (where the planar sheets of the CW-complex are bounded by arcs of confocal quadrics) such that the geodesic flow and the billiard system have the same Fomenko-Zieschang invariants.…”

“…Consider a set of domains B 0 each of which is bounded by two arcs of the ellipses with parameters b − f i and b − f i+1 and two arcs of the hyperbolae with parameters g j and g j+1 . We glue this set of mn billiards into a torus T (B 0 ) in the order of the respective edges of the piecewise linear functions f and g. The Fomenko-Zieschang invariant of the resulting graph contains the subgraphs W 2 ( f ) and W 2 ( g) and coincides with the invariant of the geodesic flow (see more details in [55]). Now we let the foci of the confocal quadric approach each other.…”

“…Integrable flows on a sphere and a torus were modelled by Vedyushkina and Fomenko [55], [46] by means of integrable circular topological billiards (for a linear first integral) and confocal topological billiards and billiard books (for a quadratic first integral). With each such flow defined by a non-trivial Riemannian metric they associated a piecewise planar table with plane metric inside 2-cells, and with these planar parts glued isometrically along their boundaries.…”