Abstract
Gromov (2001) and Sturm (2003) proved that any four points in a CAT(0) space satisfy a certain family of inequalities. We call those inequalities the ⊠-inequalities, following the notation used by Gromov. In this paper, we prove that a metric space X containing at most five points admits an isometric embedding into a CAT(0) space if and only if any four points in X satisfy the ⊠-inequalities. To prove this, we introduce a new family of necessary conditions for a metric space to admit an isometric embedding into a CAT(0) space by modifying and generalizing Gromov’s cycle conditions. Furthermore, we prove that if a metric space satisfies all those necessary conditions, then it admits an isometric embedding into a CAT(0) space. This work presents a new approach to characterizing those metric spaces that admit an isometric embedding into a CAT(0) space.
References
[1] S. Alexander, V. Kapovitch, and A. Petrunin. Alexandrov meets Kirszbraun. Proceedings of the Gökova Geometry-Topology Conference 2010. Int. Press, Somerville, MA, 2011, 88–109.Search in Google Scholar
[2] A. Andoni, A. Naor, and O. Neiman. Snowflake universality of Wasserstein spaces. Ann. Sci. Éc. Norm. Supér. (4), 51(3):657–700, 2018.10.24033/asens.2363Search in Google Scholar
[3] I. D. Berg and I. G. Nikolaev. Quasilinearization and curvature of Aleksandrov spaces. Geom. Dedicata, 133:195–218, 2008.10.1007/s10711-008-9243-3Search in Google Scholar
[4] M. R. Bridson and A. Haefliger. Metric spaces of non-positive curvature, volume 319 of Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences]. Springer-Verlag, Berlin, 1999.10.1007/978-3-662-12494-9Search in Google Scholar
[5] D. Burago, Y. Burago, and S. Ivanov. A course in metric geometry, volume 33 of Graduate Studies in Mathematics. American Mathematical Society, Providence, RI, 2001.10.1090/gsm/033Search in Google Scholar
[6] Y. Burago, M. Gromov, and G. Perelman. A. D. Aleksandrov spaces with curvatures bounded below. Uspekhi Mat. Nauk, 47(2(284)):3–51, 222, 1992.10.1070/RM1992v047n02ABEH000877Search in Google Scholar
[7] P. Enflo. On the nonexistence of uniform homeomorphisms between Lp-spaces. Ark. Mat., 8:103–105, 1969.10.1007/BF02589549Search in Google Scholar
[8] A. Eskenazis, M. Mendel, and A. Naor. Nonpositive curvature is not coarsely universal. Invent. Math., 217(3):833–886, 2019.10.1007/s00222-019-00878-1Search in Google Scholar
[9] M. Gromov. Metric structures for Riemannian and non-Riemannian spaces, volume 152 of Progress in Mathematics. Birkhäuser Boston, Inc., Boston, MA, 1999. Based on the 1981 French original [ MR0682063 (85e:53051)], With appendices by M. Katz, P. Pansu and S. Semmes, Translated from the French by Sean Michael Bates.Search in Google Scholar
[10] M. Gromov. CAT(κ)-spaces: construction and concentration. Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov. (POMI), 280(Geom. i Topol. 7):100–140, 299–300, 2001.Search in Google Scholar
[11] B. Kleiner and B. Leeb. Rigidity of quasi-isometries for symmetric spaces and Euclidean buildings. Inst. Hautes Études Sci. Publ. Math., (86):115–197 (1998), 1997.10.1007/BF02698902Search in Google Scholar
[12] T. Kondo, T. Toyoda, and T. Uehara. On a question of Gromov about the Wirtinger inequalities. Geom. Dedicata, 195(1):203–214, 2018.10.1007/s10711-017-0284-3Search in Google Scholar
[13] U. Lang and V. Schroeder. Kirszbraun’s theorem and metric spaces of bounded curvature. Geom. Funct. Anal., 7(3):535–560, 1997.10.1007/s000390050018Search in Google Scholar
[14] N. Lebedeva, A. Petrunin and V. Zolotov. Bipolar comparison. Geom. Funct. Anal., 29(1):258–282, 2019.10.1007/s00039-019-00481-9Search in Google Scholar
[15] J. G. Rešetnjak. Non-expansive maps in a space of curvature no greater than K. Sibirsk. Mat. Ž, 9:918–927, 1968.10.1007/BF02199105Search in Google Scholar
[16] T. Sato. An alternative proof of Berg and Nikolaev’s characterization of CAT(0)-spaces via quadrilateral inequality. Arch. Math. (Basel), 93(5):487–490, 2009.10.1007/s00013-009-0057-9Search in Google Scholar
[17] K.-T. Sturm. Probability measures on metric spaces of nonpositive curvature. In Heat kernels and analysis on manifolds, graphs, and metric spaces (Paris, 2002), volume 338 of Contemp. Math., pages 357–390. Amer. Math. Soc., Providence, RI, 2003.10.1090/conm/338/06080Search in Google Scholar
[18] T. Toyoda. A non-geodesic analogue of Reshetnyak’s majorization theorem. Preprint available at https://arxiv.org/abs/1907.09067, 2019.Search in Google Scholar
© 2020 Tetsu Toyoda, published by De Gruyter
This work is licensed under the Creative Commons Attribution 4.0 International License.
