Two theorems in the mathematical field of Riemannian geometry bear the name Myers–Steenrod theorem, both from a 1939 paper by Myers and Steenrod. The first states that every distance-preserving map (that is, an isometry of metric spaces) between two connected Riemannian manifolds is a smooth isometry of Riemannian manifolds. A simpler proof was subsequently given by Richard Palais in 1957. The main difficulty lies in showing that a distance-preserving map, which is a priori only continuous, is actually differentiable.
The second theorem, which is much more difficult to prove, states that the isometry group of a Riemannian manifold is a Lie group. For instance, the group of isometries of the two-dimensional unit sphere is the orthogonal group O(3).
YouTube Encyclopedic
-
1/3Views:4442 772581
-
Haynes Miller: Some homological localization theorems
-
Marc Levine - "The Motivic Fundamental Group"
-
Marc Levine: The rational motivic sphere spectrum and motivic Serre finiteness
Transcription
References
- Myers, S. B.; Steenrod, N. E. (1939), "The group of isometries of a Riemannian manifold", Ann. of Math., 2, 40 (2): 400–416, doi:10.2307/1968928, JSTOR 1968928
- Palais, R. S. (1957), "On the differentiability of isometries", Proceedings of the American Mathematical Society, 8 (4): 805–807, doi:10.1090/S0002-9939-1957-0088000-X
 | This Riemannian geometry-related article is a stub. You can help Wikipedia by expanding it. |
This page was last edited on 12 August 2023, at 00:26