Geometry & Topology

Blocking light in compact Riemannian manifolds

Jean-François Lafont and Benjamin Schmidt

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We study compact Riemannian manifolds (M,g) for which the light from any given point xM can be shaded away from any other point yM 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.

Article information

Geom. Topol., Volume 11, Number 2 (2007), 867-887.

Received: 3 August 2006
Accepted: 21 March 2007
First available in Project Euclid: 20 December 2017

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 53C22: Geodesics [See also 58E10]
Secondary: 53C20: Global Riemannian geometry, including pinching [See also 31C12, 58B20] 53B20: Local Riemannian geometry

Riemannian manifold geodesic blocking light flat manifold Euclidean building


Lafont, Jean-François; Schmidt, Benjamin. Blocking light in compact Riemannian manifolds. Geom. Topol. 11 (2007), no. 2, 867--887. doi:10.2140/gt.2007.11.867.

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