Gutkin billiard tables in higher dimensions and rigidity

TL;DR

This paper investigates Gutkin billiard tables in higher dimensions, proving rigidity results that characterize spheres and special geometries, and introduces a new generating function for billiards in multiple dimensions.

Contribution

It extends Gutkin's results to higher dimensions, providing a classification and a new generating function for billiards in these spaces.

Findings

Only spheres have the constant angle invariant curve in 3D.

Higher-dimensional Gutkin billiard tables are either spheres or have special geometry.

A new generating function for billiards in higher dimensions is derived.

Abstract

E. Gutkin found a remarkable class of convex billiard tables in the plane which have a constant angle invariant curve. In this paper we prove that in dimension 3 only round sphere has such a property. For dimension greater than 3 it must be either a sphere or to have a very special geometric properties. In 2-dimensional case we prove a rigidity result for Gutkin billiard tables. This is done with the help of a new generating function introduced recently for billiards in our joint paper with A.E. Mironov. A formula for this generating function in higher dimensions is found.