Abstract
We consider inner metric spaces of curvature bounded below in the sense of Wald, without assuming local compactness or existence of minimal curves. We first extend the Hopf-Rinow theorem by proving existence, uniqueness, and “almost extendability” of minimal curves from any point to a denseG δ subset. An immediate consequence is that Alexandrov’s comparisons are meaningful in this setting. We then prove Toponogov’s theorem in this generality, and a rigidity theorem which characterizes spheres. Finally, we use our characterization to show the existence of spheres in the space of directions at points in a denseG δ set. This allows us to define a notion of “local dimension” of the space using the dimension of such spheres. If the local dimension is finite, the space is an Alexandrov space.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
Atkin, C. J., The Hopf-Rinow theorem is false in infinite dimensions.Bull. London Math. Soc. 7, 261–266 (1975).
Berestovskii, V. N. Spaces with bounded curvature and distance geometry.Siberian Math. J. 27, 11–25 (1986).
Burago, yu., Gromov, M., and Perelman, G. Alexandrov’s spaces with curvatures bounded from below I, preprint.
Burago, Yu., Gromov, M., and Perelman, G. Alexandrov’s spaces with curvatures bounded from below I (revised).Russian Math. Surveys 47, 1–58 (1992).
Cheeger, J., and Ebin, D.Comparison Theorems in Riemannian Geometry, North-Holland, Amsterdam, 1975.
Cohn-Vossen, S. Existenz Kurzester Wege.Doklady SSSR 8, 339–342 (1935).
Ekeland, Ivar, The Hopf-Rinow theorem in infinite dimension.J. Differential Geom. 13, 287–301 (1978).
Gromov, M. Groups of polynomial growth and expanding maps.Publ. Math. I.H.E.S. 53, 53–78 (1981).
Grossman, N. Hilbert manifolds without epiconjugate points.Proc. Amer. Math. Soc. 16, 1365–1371 (1965).
Grove, K. and Petersen, P. Manifolds near the boundary of existence.J. Differential Geom. 33, 379–394 (1991).
Grove, K. and Petersen, P. On the excess of metric spaces and manifolds, unpublished.
Nagata, J.,Modern Dimension Theory, Heldermann, Berlin, 1983.
Otsu, Y., and Shioya, The Riemannian structure of Alexandrov spaces, preprint.
Otsu, Y., Shiohama, K., and Yamaguchi, T. A new version of differentiable sphere theorem,Invent. Math. 98, 219–228 (1989).
Plaut, C., Almost Riemannian spaces.J. Differential Geom. 34, 515–537 (1991).
Plaut, C., A metric characterization of manifolds with boundary.Compositio Math. 81, 337–354 (1992).
Plaut, C. Metric curvature, convergence, and topological finiteness.Duke Math. J. 66, 43–57 (1992).
Rinow, W.,Die Innere Geometrie der Metrischen Räume, Springer-Verlag, Berlin, 1961.
Shiohama, K., and Yamaguchi, T. Positively curved manifolds with restricted diameters. InGeometry of Manifolds, pp. 345–350, Academic Press, 1989.
Wald, A., Bergründung einer koordinatenlosen Differentialgeometrie der Flächen.Ergebnisse eines mathematischen Kolloquiums 7, 24–46 (1935).
Yamaguchi, T. Collapsing and pinching under a lower curvature bound.Ann. of Math. 133, 317–357 (1991).
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Plaut, C. Spaces of Wald-Berestovskii curvature bounded below. J Geom Anal 6, 113–134 (1996). https://doi.org/10.1007/BF02921569
Received:
Issue Date:
DOI: https://doi.org/10.1007/BF02921569