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• CommentRowNumber1.
• CommentAuthorDmitri Pavlov
• CommentTimeDec 2nd 2022

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## Description

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.

## References

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.
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