The Journal of Geometric Analysis

, Volume 6, Issue 1, pp 113–134

Spaces of Wald-Berestovskii curvature bounded below


DOI: 10.1007/BF02921569

Cite this article as:
Plaut, C. J Geom Anal (1996) 6: 113. doi:10.1007/BF02921569


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 denseGδ 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 denseGδ 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.

Math Subject Classification

53C20 53C70 

Key Words and Phrases

Wald curvature Hopf-Rinow theorem Toponogov’s theorem Alexandrov space 

Copyright information

© Mathematica Josephina, Inc. 1996

Authors and Affiliations

  1. 1.University of TennesseeKnoxville

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