Myers–Steenrod theorem

Template:Short description 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 $\mathrm{I}\mathrm{s}\mathrm{o}\mathrm{m}(M)$ of a connected ${𝒞}^2$ Riemannian manifold $M$ is a Lie group in a way that is compatible with the compact-open topology and such that the action $\mathrm{I}\mathrm{s}\mathrm{o}\mathrm{m}(M)\times M⟶M$ is ${𝒞}^1$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 $\mathrm{I}\mathrm{s}\mathrm{o}\mathrm{m}(M)$ is replaced by a locally compact transformation group of diffeomorphisms of $M$.

References

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