Geometry & Topology

Blocking light in compact Riemannian manifolds

Jean-François Lafont and Benjamin Schmidt

Full-text: Open access

Abstract

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

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

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

Permanent link to this document
https://projecteuclid.org/euclid.gt/1513799863

Digital Object Identifier
doi:10.2140/gt.2007.11.867

Mathematical Reviews number (MathSciNet)
MR2326937

Zentralblatt MATH identifier
1135.53028

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

Keywords
Riemannian manifold geodesic blocking light flat manifold Euclidean building

Citation

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. https://projecteuclid.org/euclid.gt/1513799863


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References

  • M Berger, Lectures on geodesics in Riemannian geometry, Tata Institute of Fundamental Research Lectures on Mathematics 33 (1965)
  • A L Besse, Manifolds all of whose geodesics are closed, Ergebnisse series 93, Springer, Berlin (1978) With appendices by D B A Epstein, J-P Bourguignon, L Bérard-Bergery, M Berger and J L Kazdan
  • K S Brown, Buildings, Springer Monographs in Mathematics, Springer, New York (1998) Reprint of the 1989 original
  • K Burns, E Gutkin, Growth of the number of geodesics between points and insecurity for Riemannian manifolds
  • M W Davis, Buildings are ${\rm CAT}(0)$, from: “Geometry and cohomology in group theory (Durham, 1994)”, London Math. Soc. Lecture Note Ser. 252, Cambridge Univ. Press (1998) 108–123
  • M P do Carmo, Riemannian geometry, Mathematics: Theory & Applications, Birkhäuser, Boston (1992) Translated from the second Portuguese edition by F Flaherty
  • D Fomin, Zadaqi Leningradskih Matematitcheskih Olimpiad (Leningrad 1990)
  • L W Green, Auf Wiedersehensflächen, Ann. of Math. $(2)$ 78 (1963) 289–299
  • E Gutkin, Blocking of billiard orbits and security for polygons and flat surfaces, Geom. Funct. Anal. 15 (2005) 83–105
  • E Gutkin, Insecure configurations in lattice translation surfaces, with applications to polygonal billiards, Discrete Contin. Dyn. Syst. 16 (2006) 367–382
  • E Gutkin, Blocking of orbits and the phenomenon of (in)security for the billiard in polygons and flat surfaces, preprint IHES/M/03/06
  • E Gutkin, V Schroeder, Connecting geodesics and security of configurations in compact locally symmetric spaces, Geom. Dedicata 118 (2006) 185–208
  • P Hiemer, V Snurnikov, Polygonal billiards with small obstacles, J. Statist. Phys. 90 (1998) 453–466
  • N D Lebedeva, Spaces without conjugate points and with fundamental group of polynomial growth, Algebra i Analiz 16 (2004) 69–81 Russian; English translation: St. Petersburg Math. J. 16 (2005) 341–348
  • R Mañé, On the topological entropy of geodesic flows, J. Differential Geom. 45 (1997) 74–93
  • T Monteil, A counter-example to the theorem of Hiemer and Snurnikov, J. Statist. Phys. 114 (2004) 1619–1623
  • T Monteil, Finite blocking versus pure periodicity
  • T Monteil, On the finite blocking property, Ann. Inst. Fourier $($Grenoble$)$ 55 (2005) 1195–1217
  • T Monteil, A homological condition for a dynamical and illuminatory classification of torus branched coverings
  • A Nabutovsky, R Rotman, The length of the shortest closed geodesic on a 2-dimensional sphere, Int. Math. Res. Not. (2002) 1211–1222
  • J B Pesin, Formulas for the entropy of the geodesic flow on a compact Riemannian manifold without conjugate points, Mat. Zametki 24 (1978) 553–570, 591 Russian; English translation: Math. Notes 24 (1978) 796-805
  • S Sabourau, Filling radius and short closed geodesics of the 2-sphere, Bull. Soc. Math. France 132 (2004) 105–136
  • L A Santaló, Integral geometry in general spaces, from: “Proceedings of the ICM (Cambridge, MA, 1950) Vol. 1”, Amer. Math. Soc. (1952) 483–489
  • J-P Serre, Homologie singulière des espaces fibrés. Applications, Ann. of Math. $(2)$ 54 (1951) 425–505
  • F W Warner, The conjugate locus of a Riemannian manifold, Amer. J. Math. 87 (1965) 575–604
  • F W Warner, Conjugate loci of constant order, Ann. of Math. $(2)$ 86 (1967) 192–212
  • O Zoll, Ueber Flächen mit Scharen geschlossener geodätischer Linien, Math. Ann. 57 (1903) 108–133