, Volume 6, Issue 1, pp 113-134

Spaces of Wald-Berestovskii curvature bounded below

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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.