A metric characterization of Riemannian spaces (Q1586989)

A. Wald and A. D. Alexandrov obtained a metric characterization of two-dimensional Riemannian manifolds. The author obtains a metric characterization of \(n\)-dimensional Riemannian manifolds: if a locally compact metric space \(M\) with intrinsic metric which admits local geodesic extensibility and the curvature of the generalized tangent bundle \(T^m(M)\) for some \(m=1,2,\dots\) is Hölder-continuous with exponent \(\alpha\in (0,1)\) then \(M\) is isometric to a \(C^{m+2,\alpha}\)-smooth Riemannian manifold.