Added:

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.

- S. B. Myers, N. E. Steenrod,
*The Group of Isometries of a Riemannian Manifold*, The Annals of Mathematics 40:2 (1939), 400–416. doi.

A simplified proof that isometries are differentiable is given by

- Richard S. Palais,
*On the differentiability of isometries*, Proceedings of the American Mathematical Society 8:4 (1956), 805–807. doi.