  We study compact Riemannian manifolds for which the light between any pair of
points is blocked by finitely many point shades. Compact flat Riemannian
manifolds are known to have this finite blocking property. We conjecture that
amongst compact Riemannian manifolds this finite blocking property
characterizes the flat metrics. Using entropy considerations, we verify this
conjecture amongst metrics with nonpositive sectional curvatures. Using the
same approach, K. Burns and E. Gutkin have independently obtained this result.
Additionally, we show that compact quotients of Euclidean buildings have the
finite blocking property.
  On the positive curvature side, we conjecture that compact Riemannian
manifolds with the same blocking properties as compact rank one symmetric
spaces are necessarily isometric to a compact rank one symmetric space. We
include some results providing evidence for this conjecture.
