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Let $X$ be locally compact Alexandrov space whose curvature satisfies both inequalities $\geq K$ and $\leq K$. What can be said about such a space? Is it locally isometric to the standard Riemannian manifold of constant curvature $K$ (e.g. sphere, Euclidean, or hyperbolic space)?

(Recall that if $X$ is a smooth Riemannian manifold then the answer is positive.)

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See Theorem 10.10.13 in the book A Course in Metric Geometry by Burago-Burago-Ivanov. (The statement is on google books here.) The theorem is attributed to I. Nikolaev but I don't have the precise original reference on me at the moment, though you could track it down through the Burago-Burago-Ivanov book.

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