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Journal of Geometry

, 109:25 | Cite as

The cone metric of a Busemann space

  • Pavel Andreev
Article

Abstract

We introduce a metric \(d_c\) on the Busemann space (Xd) such that the horofunction compactification of the space \((X, d_c)\) is equivalent to the geodesic compactification of the initial space. The space \((X, d_c)\) is not geodesic in general. It is shown that \((X, d_c)\) is geodesic if and only if X is a real tree.

Keywords

Busemann space cone metric geodesic compactification 

Mathematics Subject Classification

Primary 53C23 Secondary 53C70 

References

  1. 1.
    Andreev, P.: Semilinear metric semilattices on \(\,\mathbb{R}\) -trees. Russian Math. (Iz. VUZ) 51(6), 1–10 (2007)Google Scholar
  2. 2.
    Andreev, P.: Geometry of ideal boundaries of geodesic spaces with nonpositive curvature in the sense of Busemann. Sib. Adv. Math. 18(2), 95–102 (2008)CrossRefGoogle Scholar
  3. 3.
    Ball, B.J., Yokura, S.: Compactifications determined by subsets of \(\,C^\ast (X)\). Topology Appl. 13(1), 1–13 (1982)Google Scholar
  4. 4.
    Birkhoff, G.: Lattice Theory. American Mathematical Society, New York (1940)CrossRefzbMATHGoogle Scholar
  5. 5.
    Bridson, M.R., Haefliger, A.: Metric Spaces of Non-positive Curvature, Grundlehren Math. Wiss., vol. 319. Springer, Berlin (1999)CrossRefzbMATHGoogle Scholar
  6. 6.
    Busemann, H.: The Geometry of Geodesics. Academic Press Inc., New York (1955)zbMATHGoogle Scholar
  7. 7.
    Chandler, R.E.: Hausdorff compactifications. Lecture Notes in Pure and Applied Mathematics, vol. 23. Marcel Dekker Inc, New York (1976)Google Scholar
  8. 8.
    Gillman, L., Jerison, M.: Rings of continuous functions. The University Series in Higher Mathematics, D. Van Nostrand Co., Inc., Princeton (1960)Google Scholar
  9. 9.
    Gromov, M.: Hyperbolic manifolds, groups and actions. In: Riemann surfaces and related topics: Proceedings of the 1978 Stony Brook Conference (State Univ. New York, Stony Brook, NY, 1978), pp. 183–213, Ann. Math. Stud., 97, Princeton University Press, Princeton, NJGoogle Scholar
  10. 10.
    Hotchkiss, P.K.: The boundary of a Busemann space. Proc. Am. Math. Soc. 125(7), 1903–1912 (1997)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Papadopoulos, A.: Metric spaces, convexity and non-positive curvature. Second edition, IRMA Lectures in Mathematics and Theoretical Physics, 6, EMS, Zürich (2014)Google Scholar
  12. 12.
    Rieffel, M.: Group \(\,C^\ast \) -algebras as compact quantum metric spaces. Doc. Math. 7, 605–651 (2002)Google Scholar
  13. 13.
    Walsh, C.: Horofunction boundary of finite-dimensional normed space. Math Proc. Camb. Philos. Soc. 142(3), 497–507 (2007)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer International Publishing AG, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Northern (Arctic) Federal UniversityArkhangelskRussia

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