We study compact Riemannian manifolds
for which the light
from any given point
can be shaded away from any other point
by finitely many
point shades in .
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.
Keywords
Riemannian manifold, geodesic, blocking light, flat
manifold, Euclidean building