Volume 11, issue 2 (2007)

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Blocking light in compact Riemannian manifolds

Jean-François Lafont and Benjamin Schmidt

Geometry & Topology 11 (2007) 867–887

arXiv: math.DG/0607789


We study compact Riemannian manifolds (M,g) for which the light from any given point x M can be shaded away from any other point y M by finitely many point shades in M. 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.

Riemannian manifold, geodesic, blocking light, flat manifold, Euclidean building
Mathematical Subject Classification 2000
Primary: 53C22
Secondary: 53C20, 53B20
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Received: 3 August 2006
Accepted: 21 March 2007
Published: 27 May 2007
Proposed: Benson Farb
Seconded: Walter Neumann, Tobias Colding
Jean-François Lafont
Department of Mathematics
The Ohio State University
Columbus, OH 43210
Benjamin Schmidt
Department of Mathematics
University of Chicago
5734 S University Avenue
Chicago, IL 60637