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
is a Lie group in a way that is compatible with the compact-open topology and such that the action
is a Riemannian symmetric space: for instance, the group of isometries of the two-dimensional unit sphere is the orthogonal group
is replaced by a locally compact transformation group of diffeomorphisms of