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In their 1939 paper, Myers and Steenrod proved two theorems on Riemannian manifolds.
The first theorem shows that distance-preserving maps between Riemannian manifolds are differentiable isometries.
In 1956 Palais simplified the proof and extended the result to show that a Riemannian manifold can be reconstructed from its metric space. That is to say, the functor that sends Riemannian manifolds and isometries to metric spaces and isometries is a fully faithful functor.
The second theorem proves that the group of isometries of a Riemannian manifold is a Lie group.
A simplified proof that isometries are differentiable is given by
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