I am looking for a reference to the following simple statement; it must be classical. (It is easy to proof, but I want to have a reference.)



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> A metric space $X$ that corresponds to a Riemannian manifold $(M,g)$ completely determines the underlying smooth manifold $M$ and the metric tensor $g$.
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Riemannian manifold as a metric space
Isn't this the [Myers-Steenrod theorem](https://en.wikipedia.org/wiki/Myers%E2%80%93Steenrod_theorem)? "If $(M,g)$ and $(N,h)$ are connected Riemannian manifolds and $f:(M,d\_g)\to(N,d\_h)$ is an isometry, then $f:(M,g)\to(N,h)$ is a smooth isometry"


