Ridigity of Horospherical Billiards

Abstract Let $(M,g)$ be a compact, $C^{\infty }$ boundaryless surface without focal points and genus greater than one. We define a horospherical billiard in the universal covering of the surface and we propose two notions of integrability for the billiard. We show that under either of these notions of integrability the surface has constant negative curvature.