Geometriae Dedicata

, Volume 172, Issue 1, pp 1–45 | Cite as

Möbius characterization of the boundary at infinity of rank one symmetric spaces

  • Sergei BuyaloEmail author
  • Viktor Schroeder
Original Paper


Möbius structure (on a set \(X\)) is a class of metrics having the same cross-ratios. A Möbius structure is Ptolemaic if it is invariant under inversion operations. The boundary at infinity of a \(\mathrm{CAT }(-1)\) space is in a natural way a Möbius space, which is Ptolemaic. We give a free of classification proof of the following result that characterizes the rank one symmetric spaces of noncompact type purely in terms of their Möbius geometry: Let \(X\) be a compact Ptolemy space which contains a Ptolemy circle and allows many space inversions. Then \(X\) is Möbius equivalent to the boundary at infinity of a rank one symmetric space.


Rank one symmetric spaces Möbius structure Ptolemy spaces 

Mathematics Subject Classification

53C35 53C24 



We are thankful to L. Kramer for informing us about 2-transitive group actions. The first author is also grateful to the University of Zürich for hospitality and support. We thank the referee for his or her very helpful comments and suggestions.


  1. 1.
    Buckley, S., Falk, K., Wraith, D.: Ptolemaic spaces and CAT(0). Glasg. Math. J. 51(2), 301–314 (2009)CrossRefzbMATHMathSciNetGoogle Scholar
  2. 2.
    Cowling, M., Dooley, A., Korányi, A., Ricci, F.: An approach to symmetric spaces of rank one via groups of Heisenberg type. J. Geom. Anal. 8, 199–237 (1998)CrossRefzbMATHMathSciNetGoogle Scholar
  3. 3.
    Foertsch, T., Lytchak, A., Schroeder, V.: Nonpositive curvature and the Ptolemy inequality. Int. Math. Res. Not. IMRN 22, 15 (2007). Art. ID rnm100Google Scholar
  4. 4.
    Foertsch, T., Schroeder, V.: Hyperbolicity, \(\text{ CAT }(-1)\)-spaces and Ptolemy inequality. Math. Ann. 350(2), 339–356 (2011)CrossRefzbMATHMathSciNetGoogle Scholar
  5. 5.
    Foertsch, T., Schroeder, V.: A Möbius Characterization of Metric Spheres, arXiv:math/1008.3250 (2010), to appear in Manuscripta MathGoogle Scholar
  6. 6.
    Heintze, E.: On homogeneous manifolds of negative curvature. Math. Ann. 211, 23–34 (1974)CrossRefzbMATHMathSciNetGoogle Scholar
  7. 7.
    Hiriart-Urruty, J.-B., Lemaréchal, C.: Fundamentals of Convex Analysis. Springer, Berlin (2001)CrossRefzbMATHGoogle Scholar
  8. 8.
    Kay, D.: Ptolemaic metric spaces and the characterization of geodesics by vanishing metric curvature, Ph.D. thesis, Michigan State Univ., East Lansing, MI (1963)Google Scholar
  9. 9.
    Kramer, L.: Two-transitive Lie groups. J. Reine Angew. Math. 563, 83–113 (2003)zbMATHMathSciNetGoogle Scholar
  10. 10.
    Schoenberg, I.: A remark on M. M. Day’s characterization of inner-product spaces and a conjecture of L. M. Blumenthal. Proc. Am. Math. Soc. 3, 961–964 (1952)zbMATHMathSciNetGoogle Scholar
  11. 11.
    Siebert, E.: Contractive automorphisms on locally compact groups. Mathematische Zeitschrift 191, 73–90 (1986)CrossRefzbMATHMathSciNetGoogle Scholar

Copyright information

© Springer Science+Business Media Dordrecht 2013

Authors and Affiliations

  1. 1.St. Petersburg Department of SteklovMathematical Institute RASSt. PetersburgRussia
  2. 2.Institut für MathematikUniversität ZürichZurichSwitzerland

Personalised recommendations