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 ≠ 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 ∈ 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.