Skip to Main content Skip to Navigation
Journal articles

On commuting billiards in higher-dimensional spaces of constant curvature

Alexey Glutsyuk 1
1 UMPA
UMPA-ENSL - Unité de Mathématiques Pures et Appliquées
Abstract : We consider two nested billiards in ℝd, d≥3, with C2-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.
Document type :
Journal articles
Complete list of metadatas

Cited literature [22 references]  Display  Hide  Download

https://hal-ens-lyon.archives-ouvertes.fr/ensl-01964943
Contributor : Alexey Glutsyuk <>
Submitted on : Wednesday, November 20, 2019 - 11:28:40 AM
Last modification on : Thursday, March 5, 2020 - 3:32:33 PM

File

commute-caust.pdf
Files produced by the author(s)

Identifiers

  • HAL Id : ensl-01964943, version 2
  • ARXIV : 1807.10567

Collections

Citation

Alexey Glutsyuk. On commuting billiards in higher-dimensional spaces of constant curvature. Pacific Journal of Mathematics, Mathematical Sciences Publishers, In press. ⟨ensl-01964943v2⟩

Share

Metrics

Record views

20

Files downloads

35