Blocking light in compact Riemannian manifolds

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

Keywords

Riemannian manifold, geodesic, blocking light, flat manifold, Euclidean building

Mathematical Subject Classification 2000

Primary: 53C22

Secondary: 53C20, 53B20

References

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Publication

Received: 3 August 2006

Accepted: 21 March 2007

Published: 27 May 2007

Proposed: Benson Farb

Seconded: Walter Neumann, Tobias Colding

Authors

Jean-François Lafont
Department of Mathematics The Ohio State University Columbus, OH 43210 USA

Benjamin Schmidt
Department of Mathematics University of Chicago 5734 S University Avenue Chicago, IL 60637 USA

Volume 11, issue 2 (2007)

Jean-François Lafont and Benjamin Schmidt

Abstract