Manuscripta Mathematica

, Volume 140, Issue 3–4, pp 613–620 | Cite as

Metric Möbius geometry and a characterization of spheres

  • Thomas Foertsch
  • Viktor SchroederEmail author


We obtain a Möbius characterization of the n-dimensional spheres S n endowed with the chordal metric d 0. We show that every compact extended Ptolemy metric space with the property that every three points are contained in a circle is Möbius equivalent to (S n , d 0) for some n ≥ 1.

Mathematics Subject Classification



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  1. 1.
    Bourdon M.: Structure conforme au bord et flot géodésique d’un CAT(−1)-espace. Enseign. Math. (2) 41(1-2), 63–102 (1995)MathSciNetzbMATHGoogle Scholar
  2. 2.
    Buckley S.M., Falk K., Wraith D.J.: Ptolemaic spaces and CAT(0). Glasg. J. Math. 51, 301–314 (2009)MathSciNetzbMATHCrossRefGoogle Scholar
  3. 3.
    Buyalo, S., Schroeder, V.: Möbius structures and ptolemy spaces: boundary at infinity of complex hyperbolic spaces, arXiv:1012.1699 , 2010Google Scholar
  4. 4.
    Foertsch, Th., Lytchak, A., Schroeder, V.: Nonpositive curvature and the ptolemy inequality, vol. 22, p. 15. International Mathematics Research Notices (IMRN) (2007)Google Scholar
  5. 5.
    Foertsch Th., Schroeder V.: Hyperbolicity, CAT(−1)-spaces and the ptolemy inequality. Math. Ann. 350(2), 339356 (2011)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Foertsch Th., Schroeder V.: Group actions on geodesic ptolemy spaces. Trans. Amer. Math. Soc. 363(6), 28912906 (2011)MathSciNetCrossRefGoogle Scholar
  7. 7.
    Hamenstädt U.: A geometric characterization of negatively curved locally symmetric spaces. J. Differ. Geom. 34(1), 193–221 (1991)zbMATHGoogle Scholar
  8. 8.
    Hitzelberger P., Lytchak A.: Spaces with many affine functions. Proc. AMS 135(7), 2263–2271 (2007)MathSciNetzbMATHCrossRefGoogle Scholar
  9. 9.
    Ibragimov Z.: Hyperbolizing hyperspaces. Mich. Math. J. 60(1), 215–239 (2011)zbMATHCrossRefGoogle Scholar
  10. 10.
    Schoenberg I.J.: A remark on M. M. Day’s characterization of inner-product spaces and a conjecture of L. M. Blumenthal. Proc. Amer. Math. Soc. 3, 961–964 (1952)MathSciNetzbMATHGoogle Scholar

Copyright information

© Springer-Verlag 2012

Authors and Affiliations

  1. 1.Institut für MathematikUniversität ZürichZürichSwitzerland

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