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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. Has there been any other progress? And are there other references?

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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". 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.

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