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

I

s

o

m

(

M

)

{\textstyle \mathrm {Isom} (M)}

of a connected

C

2

{\displaystyle {\mathcal {C}}^{2}}

Riemannian manifold

M

{\textstyle M}

is a Lie group in a way that is compatible with the compact-open topology and such that the action

I

s

o

m

(

M

)

×

M

⟶

M

{\textstyle \mathrm {Isom} (M)\times M\longrightarrow M}

is

C

1

{\textstyle {\mathcal {C}}^{1}}

differentiable (in both variables). This is a generalization of the easier, similar statement when

M

{\textstyle M}

is a Riemannian symmetric space: for instance, the group of isometries of the two-dimensional unit sphere is the orthogonal group

O

(

3

)

{\textstyle O(3)}

. A harder generalization is given by the Bochner-Montgomery theorem, where

I

s

o

m

(

M

)

{\textstyle \mathrm {Isom} (M)}

is replaced by a locally compact transformation group of diffeomorphisms of

M

{\textstyle M}

.

References

Myers, S. B.; Steenrod, N. E. (1939), "The group of isometries of a Riemannian manifold", Ann.