Alexey Glutsyuk

On curves with Poritsky property and Liouville net

To each planar convex closed curve C the string construction associates a family of bigger closed curves G(t) whose billiards have C as a caustic. Each curve G(t) induces a dynamical system on C: given two tangent lines to C through the same point in G(t), the tangency point of the left tangent line is sent to the tangency point of the right tangent line. A curve C is said to have Poritsky property, if this dynamical system has an invariant length element on C that is the same for all t. Curves with Poritsky property are closely related to integrable billiards. H.Poritsky have shown that the only planar curves with Poritsky property are conics. We extend his result to surfaces of constant curvature. We consider curves with Poritsky property on arbitrary Riemannian surface and obtain the following results: 1) a formula for the invariant length element in terms of the geodesic curvature and the natural length parameter; 2) each germ of curve with Poritsky property is completely determined by its 4-jet.

We also present the following very recent joint result with Sergei Tabachnikov and Ivan Izmestiev. Given a curve C bounding a topological disc on a Riemannian surface. Then the following statements are equivalent:

1) The curve C has Poritsky property.

2) The corresponding family of string constructions has Graves property: smaller curves are caustics for bigger curves.

3) The metric on the concave side of C is Liouville.

XXXVIII Workshop on Geometric Methods in Physics	30.06-6.07.2019
VIII School on Geometry and Physics	24-28.06.2019

Event sponsored by:
University of Bialystok