On commuting billiards in higher-dimensional spaces of constant curvature

Alexey Glutsyuk

Vol. 305 (2020), No. 2, 577–595

DOI: 10.2140/pjm.2020.305.577

Abstract

We consider two nested billiards in ${ℝ}^d$, $d\ge 3$, 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 which says 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).

Keywords

commuting billiards, caustics, space forms, confocal quadrics

Mathematical Subject Classification 2010

Primary: 70H99

Milestones

Received: 24 August 2018

Revised: 23 August 2019

Accepted: 3 September 2019

Published: 29 April 2020