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A. D. Alexandrov spaces with curvature bounded below. (English. Russian original) Zbl 0802.53018
Russ. Math. Surv. 47, No. 2, 1-58 (1992); translation from Usp. Mat. Nauk 47, No. 2(284), 3-51 (1992).
The theory of (basically finite-dimensional) metric spaces with curvature (in the sense of Alexandrov) bounded below is developed. Roughly speaking, it is dealing with spaces with an intrinsic metric, for which the conclusion of Toponogov’s angle comparison theorem is true (although only in the small). These spaces are defined axiomatically by their local geometric properties, without the techniques of analysis. They may have metric and topological singularities, in particular, they may not be manifolds. The class considered includes all limit spaces of sequences of complete Riemannian manifolds with sectional curvature uniformly bounded below. Alexandrov spaces arise naturally if Riemannian manifolds are considered from the viewpoint of synthetic geometry and one avoids the excessive assumptions of smoothness connected with the use of analytic techniques. Spaces with singularities may appear as limits of sequences of ordinary Riemannian manifolds, and therefore the first are necessary for studying the latter.

MSC:
53C45 Global surface theory (convex surfaces à la A. D. Aleksandrov)
53C23 Global geometric and topological methods (à la Gromov); differential geometric analysis on metric spaces
58A03 Topos-theoretic approach to differentiable manifolds
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