Hostname: page-component-cd9895bd7-gxg78 Total loading time: 0 Render date: 2024-12-26T19:41:20.006Z Has data issue: false hasContentIssue false
  • English
  • Français

PTOLEMAIC SPACES AND CAT(0)

Published online by Cambridge University Press:  01 May 2009

S. M. BUCKLEY ,
K. FALK  and
D. J. WRAITH
S. M. BUCKLEY
Affiliation:
Department of Mathematics, National University of Ireland Maynooth, Co. Kildare, Ireland e-mail: [email protected]; [email protected]; [email protected]
K. FALK
Affiliation:
Department of Mathematics, National University of Ireland Maynooth, Co. Kildare, Ireland e-mail: [email protected]; [email protected]; [email protected]
D. J. WRAITH
Affiliation:
Department of Mathematics, National University of Ireland Maynooth, Co. Kildare, Ireland e-mail: [email protected]; [email protected]; [email protected]
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

We consider Ptolemy's inequality in a metric space setting. It is not hard to see that CAT(0) spaces satisfy this inequality. Although the converse is not true in full generality, we show that if our Ptolemaic space is either a Riemannian or Finsler manifold, then it must also be CAT(0). Ptolemy's inequality is closely related to inversions of metric spaces. We exploit this link to establish a new characterization of Euclidean space amongst all Riemannian manifolds.

Keywords

53C2053C6051F99
Type
Research Article
Information
Glasgow Mathematical Journal , Volume 51 , Issue 2 , May 2009 , pp. 301 - 314
Copyright
Copyright © Glasgow Mathematical Journal Trust 2009

References

REFERENCES

1.Aseev, V. V., Sychëv, A. S. and Tetenov, A. V., Möbius-invariant metrics and generalized angles in Ptolemaic spaces, Siberian Math. J. 46 (2005) 189–204.CrossRefGoogle Scholar
2.Bao, D., Chern, S. S. and Shen, Z., An introduction to Riemann–Finsler geometry (Springer Verlag, New York, 2000).CrossRefGoogle Scholar
3.Berger, M., A panoramic view of riemannian geometry (Springer Verlag, Berlin, Heidelberg, New York, 2003).CrossRefGoogle Scholar
4.Blumenthal, L. M., Remarks on a weak four-point property, Revista Ci. Lima 45 (1943), 183–193.Google Scholar
5.Blumenthal, L. M., Theory and applications of distance geometry, 2nd Ed. (Chelsea Publishing Co., New York, 1970).Google Scholar
6.Bridson, M. R. and Haefliger, A., Metric spaces of non-positive curvature (Springer Verlag, Berlin, 1999).CrossRefGoogle Scholar
7.Buckley, S. M., Herron, D. and Xie, X., Metric space inversions, quasihyperbolic distance, and uniform spaces, Preprint (2006).Google Scholar
8.Burago, D., Burago, Y. and Ivanov, S., A course in metric geometry, in Graduate studies in mathematics, vol. 33 (AMS, 2001).CrossRefGoogle Scholar
9.Foertsch, T., Lytchak, A. and Schroeder, V., Nonpositive curvature and the Ptolemy inequality, Int. Math. Res. Not. 2007 (2007), article ID rnm100.Google Scholar
10.Foertsch, T. and Schroeder, V., Hyperbolicity CAT(-1)-spaces and the Ptolemy inequality, Preprint (2006).Google Scholar
11.Fukaya, K., Hausdorff convergence of Riemannian manifolds and its applications, Adv. Stud. Pure Math. 18-I (1990), 143–238.CrossRefGoogle Scholar
12.Gromov, M., Metric structures for Riemannian and non-Riemannian spaces, in Progress in Mathematics vol. 152 (Birkhäuser, 2001).Google Scholar
13.Kay, D. C., The ptolemaic inequality in Hilbert geometries, Pacific J. Math. 21 (1967), 293–301.CrossRefGoogle Scholar
14.Klamkin, M. S. and Meir, A., Ptolemy's inequality, chordal metric, multiplicative metric, Pacific J. Math. 101 (1982) 389–392.CrossRefGoogle Scholar
15.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 (1952) 961–964.Google Scholar
16.Whitehead, I. J., Convex regions in the geometry of paths, Quart. J. Math. Oxford 3 (1932) 33–42.CrossRefGoogle Scholar