Alexandrov space

An Alexandrov space is a generalization of a Riemannian manifold in which curvature bounds are defined not through differential equations but through metric triangle comparison — a space has curvature bounded below by κ if every geodesic triangle is at least as thick as its comparison triangle in the model space of constant curvature κ. Named after the Russian mathematician Alexandr Alexandrov, who developed the theory for convex surfaces in the 1940s, these spaces retain rich geometric structure — including a well-defined notion of tangent cone, stratification by dimension, and topological regularity theorems — despite potentially lacking any smooth structure whatsoever. Alexandrov spaces emerge naturally as Gromov-Hausdorff limits of sequences of Riemannian manifolds with uniform curvature bounds, and they played a decisive role in Perelman's proof of the geometrization conjecture by providing the geometric framework for analyzing singularities in the Ricci flow. The theory demonstrates that curvature is not a smooth phenomenon but a metric one — a principle that has become foundational to modern metric geometry.