On commuting billiards in higher-dimensional spaces of constant curvature

TL;DR

This paper proves that in higher-dimensional spaces of constant curvature, commuting billiard reflections imply the billiards are confocal ellipsoids, solving a longstanding conjecture and extending classical theorems to curved spaces.

Contribution

It establishes that commuting billiard actions in higher dimensions are characterized by confocal ellipsoids, extending Berger's theorem to spaces of constant curvature.

Findings

Commuting billiard reflections imply confocal ellipsoids in Euclidean space.

Extension of Berger's theorem to spaces of constant curvature.

Complete solution of the Commuting Billiard Conjecture in higher dimensions.

Abstract

We consider two nested billiards in $\mathbb R^d$, $d\geq3$, with $C^2$-smooth strictly convex boundaries. We prove that if the corresponding actions by reflections on the space of oriented lines commute, then the billiards are confocal ellipsoids. This together with the previous analogous result of the author in two dimensions solves completely the Commuting Billiard Conjecture due to Sergei Tabachnikov. The main result is deduced from the classical theorem due to Marcel Berger saying that in higher dimensions only quadrics may have caustics. We also prove versions of Berger's theorem and the main result for billiards in spaces of constant curvature: space forms.