We konwn that the smooth Riemannian manifold with nonnegative curvature is Alexandrov space (with induced metric) with nonnegative curvature.

What about the converse? i.e Given a smooth metric d on smooth manifold M such that M is a Alexandrov space with nonnegative curvature, can we find a smooth Riemannian structure g on M so that d is induced by g ?

Otsu&Shioya showed partial results in the paper The Riemannian structure of Alexandrov spaces [enter link description here][1] . Has any progress and other reference? [1]: http://www.ams.org/mathscinet/search/publdoc.html?r=1&pg1=MR&s1=1274133&loc=fromrevtext

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?