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 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).

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 
• 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