Abstract
Distance functions of metric spaces with lower curvature bound, by definition, enjoy various metric inequalities; triangle comparison, quadruple comparison and the inequality of Lang–Schroeder–Sturm. The purpose of this paper is to study the extremal cases of these inequalities and to prove rigidity results. The spaces which we shall deal with here are Alexandrov spaces which possibly have infinite dimension and are not supposed to be locally compact.
Similar content being viewed by others
References
Berg I.-D., Nikolaev I.-G.: Quasilinearization and curvature of Aleksandrov spaces. Geom. Dedicata 133, 195–218 (2008)
Burago, D., Burago, Y., Ivanov, S.: A course in metric geometry. Graduate Studies in Mathematics, vol. 33. American Mathematical Society, Providence (2001)
Burago, Y., Gromov, M., Perelman, G.: A.D. Aleksandrov spaces with curvatures bounded below. Uspekhi Mat. Nauk 47(2), 3–51, 222 (1992); translation in Russian Math. Surveys 47(2), 1–58 (1992)
Benyamini, Y., Lindenstrauss, J.: Geometric Nonlinear Functional Analysis, vol. 1. American Mathematical Society Colloquium Publications, 48, Providence (2000)
Grove K., Markvorsen S.: New extremal problems for the Riemannian recognition program via Alexandrov geometry. J. Am. Math. Soc. 8(1), 1–28 (1995)
Grove K., Wilhelm F.: Hard and soft packing radius theorems. Ann. Math. 142(2), 213–237 (1995)
Halbeisen S.: On tangent cones of Alexandrov spaces with curvature bounded below. Manuscr. Math. 103(2), 169–182 (2000)
Lang U., Schroeder V.: Kirszbraun’s theorem and metric spaces of bounded curvature. Geom. Funct. Anal. 7(3), 535–560 (1997)
Lebedeva N., Petrunin A.: Curvature bounded below: a definition a la Berg-Nikolaev. Electron. Res. Announc. Math. Sci. 17, 122–124 (2010)
Loveland L.D., Valentine J.: Congruent embedding of metric quadruples on a unit sphere. J. Reine Angew. Math. 261, 205–209 (1973)
Mitsuishi A.: A splitting theorem for infinite dimensional Alexandrov spaces with nonnegative curvature and its applications. Geom. Dedicata 144, 101–114 (2010)
Morgan, J., Tian, G.: Ricci flow and the Poincaré conjecture. Clay Mathematics Monographs, vol. 3. American Mathematical Society, Providence (2007)
Ohta, S.: Barycenters in Alexandrov spaces of curvature bounded below. Adv. Geom. Preprint (2009, to appear)
Ohta S., Pichot M.: A note on Markov type constants. Arch. Math. (Basel) 92(1), 80–88 (2009)
Otsu Y., Shioya T.: The Riemannian structure of Alexandrov spaces. J. Differ. Geom. 39(3), 629–658 (1994)
Perelman, G., Petrunin, A.: Quasigeodesics and Gradient curves in Alexandrov spaces. Unpublished preprint (1995)
Petrunin, A.: Semiconcave functions in Alexandrov’s geometry. Surveys in Differential Geometry, vol. XI, pp. 137–201. International Press (2007)
Plaut C.: Spaces of Wald curvature bounded below. J. Geom. Anal. 6(1), 113–134 (1996)
Plaut C.: Metric Spaces of Curvature ≥ k. Handbook of Geometric Topology, pp. 819–898. North-Holland, Amsterdam (2002)
Sato T.: An alternative proof of Berg and Nikolaev’s characterization of CAT(0)-spaces via quadrilateral inequality. Arch. Math. (Basel) 93(5), 487–490 (2009)
Shiohama, K.: An introduction to the geometry of Alexandrov spaces. Lecture Notes Series, vol. 8. Seoul National University, Research Institute of Mathematics, Seoul (1993)
Sturm K.-T.: Metric spaces of lower bounded curvature. Expo. Math. 17(1), 35–47 (1999)
Sturm K.-T.: Monotone approximation of energy functionals for mappings into metric spaces II. Potential Anal. 11(4), 359–386 (1999)
Valentine J., Wayment S.: Metric transforms and the hyperbolic four-point property. Proc. Am. Math. Soc. 31, 232–234 (1972)
Villani, C.: Optimal transport. Old and new. Grundlehren der Mathematischen Wissenschaften, 338. Springer-Verlag, Berlin (2009)
Author information
Authors and Affiliations
Corresponding author
Additional information
T. Yokota was partially supported by Research Fellowships of the Japan Society for the Promotion of Science for Young Scientists (19.9377).
Rights and permissions
About this article
Cite this article
Yokota, T. A rigidity theorem in Alexandrov spaces with lower curvature bound. Math. Ann. 353, 305–331 (2012). https://doi.org/10.1007/s00208-011-0686-8
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00208-011-0686-8