Volume 11, issue 2 (2007)

Download this article
Download this article For screen
For printing
Recent Issues

Volume 20
Issue 4, 1807–2438
Issue 3, 1257–1806
Issue 2, 629–1255
Issue 1, 1–627

Volume 19, 6 issues

Volume 18, 5 issues

Volume 17, 5 issues

Volume 16, 4 issues

Volume 15, 4 issues

Volume 14, 5 issues

Volume 13, 5 issues

Volume 12, 5 issues

Volume 11, 4 issues

Volume 10, 4 issues

Volume 9, 4 issues

Volume 8, 3 issues

Volume 7, 2 issues

Volume 6, 2 issues

Volume 5, 2 issues

Volume 4, 1 issue

Volume 3, 1 issue

Volume 2, 1 issue

Volume 1, 1 issue

The Journal
About the Journal
Editorial Board
Editorial Interests
Editorial Procedure
Submission Guidelines
Submission Page
Subscriptions
Author Index
To Appear
Contacts
ISSN (electronic): 1364-0380
ISSN (print): 1465-3060
Blocking light in compact Riemannian manifolds

Jean-François Lafont and Benjamin Schmidt

Geometry & Topology 11 (2007) 867–887

arXiv: math.DG/0607789

Abstract

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.

Keywords
Riemannian manifold, geodesic, blocking light, flat manifold, Euclidean building
Mathematical Subject Classification 2000
Primary: 53C22
Secondary: 53C20, 53B20
References
Forward citations
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