Four equivalent properties of integrable billiards - Archive ouverte HAL Access content directly
Journal Articles Israel Journal of Mathematics Year : 2021

Four equivalent properties of integrable billiards

(1) , (2) , (3)
1
2
3
Alexey Glutsyuk
Ivan Izmestiev
  • Function : Author
  • PersonId : 842084
Serge Tabachnikov
  • Function : Author
  • PersonId : 836761

Abstract

By a classical result of Darboux, a foliation of a Riemannian surface has the Graves property (also known as the strong evolution property) if and only if the foliation comes from a Liouville net. A similar result of Blaschke says that a pair of orthogonal foliations has the Ivory property if and only if they form a Liouville net. Let us say that a geodesically convex curve on a Riemannian surface has the Poritsky property if it can be parametrized in such a way that all of its string diffeomorphisms are shifts with respect to this parameter. In 1950, Poritsky has shown that the only closed plane curves with this property are ellipses. In the present article we show that a curve on a Riemannian surface has the Poritsky property if and only if it is a coordinate curve of a Liouville net. We also recall Blaschke's derivation of the Liouville property from the Ivory property and his proof of Weihnacht's theorem: the only Liouville nets in the plane are nets of confocal conics and their degenerations. This suggests the following generalization of Birkhoff's conjecture: If an interior neighborhood of a closed geodesically convex curve on a Riemannian surface is foliated by billiard caustics, then the metric in the neighborhood is Liouville, and the curve is one of the coordinate lines.
Fichier principal
Vignette du fichier
four-prop-arx-19.pdf (1.17 Mo) Télécharger le fichier
Origin : Files produced by the author(s)
Loading...

Dates and versions

ensl-02374928 , version 1 (21-11-2019)

Identifiers

Cite

Alexey Glutsyuk, Ivan Izmestiev, Serge Tabachnikov. Four equivalent properties of integrable billiards. Israel Journal of Mathematics, 2021, 241, pp.693--719. ⟨ensl-02374928⟩
19 View
55 Download

Altmetric

Share

Gmail Facebook Twitter LinkedIn More