We know that a smooth Riemannian manifold with nonnegative curvature is an Alexandrov space (with induced metric) of nonnegative curvature.


What about the converse? That is, given a smooth metric ***d*** on a smooth manifold ***M*** such that ***M*** is an Alexandrov space with nonnegative curvature, can we find a smooth Riemannian structure ***g*** on ***M*** so that ***d*** is induced by ***g*** ?


Otsu and Shioya showed partial results in the paper [*The Riemannian structure of Alexandrov spaces*](http://www.ams.org/mathscinet/search/publdoc.html?r=1&pg1=MR&s1=1274133&loc=fromrevtext). Has there been any other progress? And are there other references?



Can we realize the smooth metric of an Alexandrov space with nonnegative curvature by a Riemannian structure?
Yes, smooth distance functions plus Alexandrov means Riemannian,
but you should make all the definitions precise.


After Otsu and Shioya, there was a paper of Perelman ["DC structure on Alexandrov space with curvature bounded below"](http://www.math.psu.edu/petrunin/papers/alexandrov/Cstructure.pdf). The key "new" ingredient is an application of the construction from the Riemann's lecture which reconstructs the metric tensor from sufficiently many distance functions. 


