#### Vol. 305, No. 2, 2020

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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
Primary: 70H99
##### Milestones
Received: 24 August 2018
Revised: 23 August 2019
Accepted: 3 September 2019
Published: 29 April 2020
##### Authors
 Alexey Glutsyuk CNRS, France (UMR 5669 (UMPA, ENS de Lyon) and UMI 2615 (Interdisciplinary Scientific Center J.-V.Poncelet)) Lyon France National Research University Higher School of Economics (HSE) Moscow Russian Federation