Four equivalent properties of integrable billiards

TL;DR

This paper establishes four equivalent characterizations of integrable billiards on Riemannian surfaces, linking geometric properties of curves and nets to integrability conditions and classical results.

Contribution

It proves that curves with the Poritsky property are coordinate lines of Liouville nets and connects this to classical properties like Graves and Ivory, extending Birkhoff's conjecture.

Findings

Poritsky property characterizes coordinate curves of Liouville nets.

Liouville nets in the plane are confocal conics and degenerations.

A neighborhood foliated by billiard caustics implies the metric is Liouville.

Abstract

By a classical result of Darboux, a foliation of a Riemannian surface has the Graves property (also known as the strong evolution property) if and only if the foliation comes from a Liouville net. A similar result of Blaschke says that a pair of orthogonal foliations has the Ivory property if and only if they form a Liouville net. Let us say that a geodesically convex curve on a Riemannian surface has the Poritsky property if it can be parametrized in such a way that all of its string diffeomorphisms are shifts with respect to this parameter. In 1950, Poritsky has shown that the only closed plane curves with this property are ellipses. In the present article we show that a curve on a Riemannian surface has the Poritsky property if and only if it is a coordinate curve of a Liouville net. We also recall Blaschke's derivation of the Liouville property from the Ivory property and his proof of Weihnacht's theorem: the only Liouville nets in the plane are nets of confocal conics and their degenerations. This suggests the following generalization of Birkhoff's conjecture: If an interior neighborhood of a closed geodesically convex curve on a Riemannian surface is foliated by billiard caustics, then the metric in the neighborhood is Liouville, and the curve is one of the coordinate lines.