Journal of Mathematical Sciences

, Volume 182, Issue 1, pp 22–36 | Cite as

Metric betweenness, Ptolemaic spaces, and isometric embeddings of pretangent spaces in ℝ

  • Viktoria V. Bilet
  • Aleksei A. Dovgoshey


We study some properties of pretangent spaces describing the infinitesimal geometry of general metric spaces. The necessary and sufficient conditions characterizing Ptolemaic pretangent spaces and pretangent spaces, where every three points are situated “on a single straight line,” are found. As a corollary, we obtain a criterion of embeddability of pretangent spaces into the set ℝ of real numbers endowed by the natural metric.


Metric spaces pretangent spaces isometric embeddings metric betweenness Ptolemaic spaces 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    F. Abdullayev, O. Dovgoshey, and M. Küçüaslan, “Compactness and boundedness of tangent spaces to metric spaces,” Beiträge Alg. Geom., 51, No. 2, 547–576 (2010).Google Scholar
  2. 2.
    F. Abdullayev, O. Dovgoshey, and M. Küçüaslan, “Metric spaces with unique pretangent spaces. Conditions of the uniqueness,” Ann. Acad. Sci. Fenn. Math., 36, No. 2, 353–392 (2011).MathSciNetMATHCrossRefGoogle Scholar
  3. 3.
    L. Ambrosio and B. Kircheim, “Currents in metric spaces,” Acta Math., 185, 1–80 (2000).MathSciNetMATHCrossRefGoogle Scholar
  4. 4.
    L. Ambrosio and B. Kircheim, “Rectifiable sets in metric and Banach spaces,” Math. Ann., 318, 527–555 (2000).MathSciNetMATHCrossRefGoogle Scholar
  5. 5.
    M. Berger, Geometry, Springer, Berlin, 1987.Google Scholar
  6. 6.
    L. M. Blumenthal, Theory and Applications of Distance Geometry, Clarendon Press, Oxford, 1953.MATHGoogle Scholar
  7. 7.
    M. Bridson and A. Haefliger, Metric Spaces of Non-Positive Curvature, Springer, Berlin, 1999.MATHGoogle Scholar
  8. 8.
    O. Dovgoshey, “Tangent spaces to metric spaces and to their subspaces,” Ukr. Mat. Visn., 5, No. 4, 470–487 (2008).MathSciNetGoogle Scholar
  9. 9.
    O. Dovgoshey and D. Dordovskyi, “Ultrametricity and metric betweenness in tangent spaces to metric spaces,” P-Adic Numbers Ultrametric Anal. Appl., 2, No. 2, 100–113 (2010).MathSciNetCrossRefGoogle Scholar
  10. 10.
    A. A. Dovgoshei and D. V. Dordovskii, “Betweenness relation and isometric imbeddings of metric spaces,” Ukr. Math. J., 61, No. 10, 1556–1567 (2009).CrossRefGoogle Scholar
  11. 11.
    O. Dovgoshey and O. Martio, Tangent Spaces to Metric Spaces, Reports in Math., 480, Helsinki Univ., Helsinki, 2008.Google Scholar
  12. 12.
    O. Dovgoshey and O. Martio, “Tangent spaces to the general metric spaces,” Rev. Roum. Math. Pures Appl., 56, No. 2, 137–155 (2011).MathSciNetGoogle Scholar
  13. 13.
    A. A. Dovgoshei and E. A. Petrov, “Ptolemaic spaces,” Sib. Mat. Zh., 52, No. 2, 283–291 (2011).MathSciNetGoogle Scholar
  14. 14.
    D. V. Dordovskyi, “Metric tangents to Euclidean spaces,” Ukr. Mat. Visn., 8, No. 2, 159–181 (2011).MathSciNetGoogle Scholar
  15. 15.
    T. Foertsch, A. Lytchak, and V. Schroeder, “Non-positive curvature and the Ptolemy inequality,” Int. Math. Res. Notice, 2007, 1–12 (2007).Google Scholar
  16. 16.
    A. Lytchak, “Differentiation in metric spaces,” Alg. Analiz, 16, 128–161 (2004).MathSciNetGoogle Scholar
  17. 17.
    K. Menger, “Untersuchungen über allgemeine Metrik. I-III,” Math. Ann., 100, 75–163 (1928).MathSciNetMATHCrossRefGoogle Scholar
  18. 18.
    V. V. Prasolov, Tasks on Planimetry [in Russian], Part I, MTsNMO, Moscow, 2006.Google Scholar
  19. 19.
    I. J. Schoenberg, “A remark on M. M. Day’s characterization of inner-product spaces and a conjecture of L. M. Blumenthal,” Proc. Amer. Soc., 3, No. 6, 961–964 (1952).MathSciNetMATHGoogle Scholar
  20. 20.
    I. J. Schoenberg, “On metric arcs of vanishing Menger curvature,” Ann. of Math., 41(2), 719–723 (1940).MathSciNetGoogle Scholar

Copyright information

© Springer Science+Business Media, Inc. 2012

Authors and Affiliations

  1. 1.Institute of Applied Mathematics and Mechanics of the NAS of UkraineDonetskUkraine

Personalised recommendations