A characterization of round spheres in terms of blocking light

Abstract

A closed Riemannian manifold $M$ is said to have cross (compact rank one symmetric space) blocking if whenever $p\ne q$ are less than the diameter apart, all light rays from $p$ can be shaded away from $q$ with at most two point shades. Similarly, a closed Riemannian manifold is said to have sphere blocking if for each $p\in M$ all the light rays from $p$ are shaded away from $p$ by a single point shade. We prove that Riemannian manifolds with cross and sphere blocking are isometric to round spheres.