Skip to main content
SpringerLink
Log in

Connecting Geodesics and Security of Configurations in Compact Locally Symmetric Spaces

  • Published:
Geometriae Dedicata Aims and scope Submit manuscript

Abstract

A pair of points in a Riemannian manifold makes a secure configuration if the totality of geodesics connecting them can be blocked by a finite set. The manifold is secure if every configuration is secure. We investigate the security of compact, locally symmetric spaces.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Bott R. and Samelson H. (1958). Applications of the theory of Morse to symmetric spaces. Amer. J. Math 80:964–1029

    Article  MathSciNet  Google Scholar 

  2. Buyalo S., Schroeder V. and Walz M. (2000). Geodesics avoiding open subsets in surfaces of negative curvature. Ergodic Theory Dynam. Systems 20:991–1006

    Article  MATH  MathSciNet  Google Scholar 

  3. Eberlein P. (1983). Euclidean de Rham factor of a lattice of nonpositive curvature. J. Differential Geom 18:209–220

    MATH  MathSciNet  Google Scholar 

  4. Eberlein P. (1996). Geometry of Nonpositively Curved Manifolds. University of Chicago Press, Chicago

    MATH  Google Scholar 

  5. Gutkin, E.: Blocking of orbits and the phenomen of (in) security for the billiard in polygons and flat surfaces, Preprint IHES/M/03/36 (2003).

  6. Gutkin E. (2005). Blocking of billiard orbits and security for polygons and flat surfaces. Geom. Funct. Anal 15: 83–105

    Article  MathSciNet  MATH  Google Scholar 

  7. Gutkin, E.: Insecure configurations in lattice translation surfaces, with applications to polygonal billiards, Preprint (2005).

  8. Gutkin E. and Judge C. (2000). Affine mappings of translation surfaces: geometry and arithmetic. Duke Math. J 103:191–213

    Article  MATH  MathSciNet  Google Scholar 

  9. Hedlund G.A. (1936). Fuchsian groups and transitive horocycles. Duke Math. J 2:530–542

    Article  MathSciNet  MATH  Google Scholar 

  10. Helgason S. (1962). Differential Geometry and Symmetric Spaces. Academic Press, New York

    MATH  Google Scholar 

  11. Helgason S. (1978). Differential Geometry, Lie Groups, and Symmetric Spaces. Academic Press, New York

    MATH  Google Scholar 

  12. Loos O. (1969). Symmetric Spaces. Benjamin, New York

    MATH  Google Scholar 

  13. Mostow, G. D.: Strong Rigidity of Locally Symmetric Spaces, Ann. Math. Stud. 78, Princeton University Press, Princeton 1973.

  14. Rauch J. (1978). Illumination of bounded domains. Amer. Math. Monthly 85:359–361

    Article  MATH  MathSciNet  Google Scholar 

  15. Veech W.A. (1977). Unique ergodicity of horospherical flows. Amer. J. Math 99:827–859

    Article  MATH  MathSciNet  Google Scholar 

  16. Veech W.A. (1992). The billiard in a regular polygon. Geom. Funct. Anal 2:341–379

    Article  MATH  MathSciNet  Google Scholar 

  17. Wolf J.A. (1963). Elliptic spaces in Grassmann manifolds. Illinois J. Math 7:447–462

    MATH  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. University of California and IMPA, Estrada Dona Castorina 110, Rio de Janeiro, Brasil, 22460–320

    Eugene Gutkin

  2. Department of Mathematics, University of Zürich, CH-8057, Zürich, Switzerland

    Viktor Schroeder

Authors
  1. Eugene Gutkin
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Viktor Schroeder
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Eugene Gutkin.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Gutkin, E., Schroeder, V. Connecting Geodesics and Security of Configurations in Compact Locally Symmetric Spaces. Geom Dedicata 118, 185–208 (2006). https://doi.org/10.1007/s10711-005-9036-x

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10711-005-9036-x

Keywords

Mathematics Subject Classification (2000)

Search

Navigation