Myers–Steenrod theorem

{{Short description|The isometry group of a Riemannian manifold is a Lie group}}

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 surjective 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 harder to prove, states that the isometry group $\backslash mathrm\{Isom\}(M)$ of a connected $\backslash mathcal\{C\}^2$ Riemannian manifold $M$ is a Lie group in a way that is compatible with the compact-open topology and such that the action $\backslash mathrm\{Isom\}(M)\backslash times\; M\; \backslash longrightarrow\; M$ is $\backslash mathcal\{C\}^1$differentiable (in both variables). This is a generalization of the easier, similar statement when $M$ is a Riemannian symmetric space: for instance, the group of isometries of the two-dimensional unit sphere is the orthogonal group $O(3)$. A harder generalization is given by the Bochner-Montgomery theorem, where $\backslash mathrm\{Isom\}(M)$ is replaced by a locally compact transformation group of diffeomorphisms of $M$.{{Cite journal |last1=Bochner |first1=Salomon |last2=Montgomery |first2=Deane |date=1946 |title=Locally Compact Groups of Differentiable Transformations |url=https://www.jstor.org/stable/1969226 |journal=Annals of Mathematics |volume=47 |issue=4 |pages=639–653 |doi=10.2307/1969226 |jstor=1969226 |issn=0003-486X}}