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Blocking light in compact
Riemannian manifolds
Jean-François Lafont and Benjamin Schmidt |
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Geometry & Topology 11 (2007) 867–887
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arXiv: math.DG/0607789 |
Abstract |
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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
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Mathematical Subject Classification 2000
Primary: 53C22
Secondary: 53C20, 53B20
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References |
Forward citations |
Publication
Received: 3 August 2006
Accepted: 21 March 2007
Published: 27 May 2007
Proposed: Benson Farb
Seconded: Walter Neumann, Tobias Colding
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Authors |
| Jean-François Lafont | |
| Department of Mathematics The Ohio State University Columbus, OH 43210 USA |
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| Benjamin Schmidt | |
| Department of Mathematics University of Chicago 5734 S University Avenue Chicago, IL 60637 USA |
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