Killing–Hopf theorem

Remarks

The idea of the proof: a theorem of Cartan says that two Riemannian manifolds of the same dimension and the same constant sectional curvature are locally isometric. For a complete simply connected $M$, a local isometry from a ball of $X_K$ extends along geodesics to a global isometry; this gives part 1. For general $M$, the universal cover with the pulled-back metric is complete, simply connected and of curvature $K$, so it is $X_K$ by part 1. The deck transformations of the covering are then isometries.

Example: the real projective space $\R P^n = S^n / \set{\pm \id}$ with the round metric is a space form of curvature $1$ with $\Gamma$ of order $2$. Flat tori $\R^n / \Z^n$ are quotients of $\R^n$ by a lattice of translations.

Completeness is needed: an open ball in $\R^n$ is flat and simply connected, but it is not isometric to $\R^n$.

Going deeper

For a student

The theorem turns the classification of space forms into group theory: find the discrete groups of isometries of $S^n$, $\R^n$ or hyperbolic space that act freely. For $K > 0$ these are finite subgroups of the orthogonal group acting freely on the sphere, which gives for example the lens spaces in dimension $3$. For $K = 0$ and $M$ compact one gets the Bieberbach groups. For $K < 0$ one gets the torsion-free discrete subgroups of the isometry group of hyperbolic space, a very rich family.