Myers–Steenrod theorem

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 ${\textstyle \mathrm {Isom} (M)}$ of a connected ${\mathcal {C}}^{2}$ Riemannian manifold ${\textstyle M}$ is a Lie group in a way that is compatible with the compact-open topology and such that the action ${\textstyle \mathrm {Isom} (M)\times M\longrightarrow M}$ is ${\textstyle {\mathcal {C}}^{1}}$differentiable (in both variables). This is a generalization of the easier, similar statement when ${\textstyle M}$ is a Riemannian symmetric space: for instance, the group of isometries of the two-dimensional unit sphere is the orthogonal group ${\textstyle O(3)}$. A harder generalization is given by the Bochner-Montgomery theorem, where ${\textstyle \mathrm {Isom} (M)}$ is replaced by a locally compact transformation group of diffeomorphisms of ${\textstyle M}$.

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