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.)



Alexandrov spaces of constant curvature
See Theorem 10.10.13 in the book A Course in Metric Geometry by Burago-Burago-Ivanov. (The statement is on google books [here](https://books.google.com/books?id=dRmIAwAAQBAJ&pg=PA404&lpg=PA404&dq=nikolaev%20upper%20curvature&source=bl&ots=NJcxT8fec8&sig=je65fJfMMACKDCz8xDSqKQE7VQc&hl=en&sa=X&ei=96kvVZ-sPMm5ggT7loGoAg&ved=0CDAQ6AEwAg#v=onepage&q=nikolaev%20upper%20curvature&f=false).) 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.


