Hopf rigidity for convex billiards on the hemisphere and hyperbolic plane (Q379543)

The author considers Hopf-type rigidity for convex billiards on surfaces with constant curvature. The main result is as follows: If \(\gamma\) is a smooth convex simple closed curve having positive geodesic curvature lying on the hemisphere or the hyperbolic plane of constant curvature of \(\pm 1\), and if no billiard configuration \(\{x_n\}\) has conjugate points, then \(\gamma\) is a geodesic circle.NEWLINENEWLINE The proof of this result use a mirror formula originally from geometric optics that is generalized to simple convex domains on Riemannian surfaces. The author reduces the claimed result to a geometric inequality that is a consequence of the isoperimetric inequality on the surface of constant curvature. The cases of the hemisphere and hyperbolic plane are handled separately.