A characterization of round spheres in terms of blocking light

TL;DR

This paper characterizes round spheres among Riemannian manifolds by introducing blocking properties related to light rays, proving that such properties uniquely identify the geometry of round spheres.

Contribution

It establishes that Riemannian manifolds with cross blocking and sphere blocking are necessarily isometric to round spheres, providing a geometric characterization.

Findings

Manifolds with cross blocking are isometric to round spheres.

Manifolds with sphere blocking are isometric to round spheres.

Blocking properties uniquely determine the spherical geometry.

Abstract

A closed Riemannian manifold is said to have cross blocking if whenever distinct points p and q are at distance less than the diameter, 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 point p, 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.