On Totally integrable magnetic billiards on constant curvature surface

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.