Myers-Steenrod theorem
= Myers-Steenrod theorem
{c}
The isometry group of a smooth connected <Riemannian manifold> is a finite-dimensional <Lie group>, and its action is smooth. An isometry is determined by its value and derivative at one point, since it preserves the exponential map and continuation along geodesics. The stabilizer embeds in $O(n)$, giving dimension at most $n(n+1)/2$. For a <compact manifold> the isometry group is compact.