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1. • CommentRowNumber1.
   • CommentAuthorDmitri Pavlov
   • CommentTimeDec 2nd 2022
   • PermaLink
   Author: Dmitri Pavlov
   Format: MarkdownItexAdded: ## 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]]. ## Related concepts * [[Riemannian manifold]] * [[Lie group]] * [[isometry]] ## References * [[S. B. Myers]], [[N. E. Steenrod]], _The Group of Isometries of a Riemannian Manifold_, The Annals of Mathematics 40:2 (1939), 400–416. [doi](http://dx.doi.org/10.2307/1968928). 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](http://dx.doi.org/10.1090/s0002-9939-1957-0088000-x). <a href="https://ncatlab.org/nlab/revision/diff/Myers-Steenrod+theorem/5">diff</a>, <a href="https://ncatlab.org/nlab/revision/Myers-Steenrod+theorem/5">v5</a>, <a href="https://ncatlab.org/nlab/show/Myers-Steenrod+theorem">current</a>

   Added:

   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.

   Related concepts

   • Riemannian manifold

   • Lie group

   • isometry

   References

   • 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.

   diff, v5, current