On Totally integrable magnetic billiards on constant curvature surface

TL;DR

This paper proves that in magnetic billiards on constant curvature surfaces, total integrability implies the boundary must be a circle, illustrating a rigidity phenomenon similar to classical billiards.

Contribution

It establishes a rigidity result for magnetic billiards, showing that total integrability constrains the boundary shape to be circular on constant curvature surfaces.

Findings

Total integrability implies boundary is a circle.

The result extends Hopf rigidity to magnetic billiards.

Supports the universality of rigidity phenomena in billiard systems.

Abstract

We consider billiard ball motion in a convex domain of a constant curvature surface influenced by the constant magnetic field. We prove that if the billiard map is totally integrable then the boundary curve is necessarily a circle. This result is a manifestation of the so-called Hopf rigidity phenomenon which was recently obtained for classical billiards on constant curvature surfaces.