Bieberbach theorem

The theorem classifies compact flat manifolds through their crystallographic covering groups.

For a compact flat Riemannian manifold ${\mathbb{R}}^n/\Gamma$, the translations in $\Gamma$ form a normal full-rank Euclidean lattice of finite index. The quotient is its finite linear holonomy. Affine conjugacy of the torsion-free crystallographic groups classifies these manifolds up to affine diffeomorphism. An abelian $\Gamma$ must consist entirely of translations: commuting with every lattice translation forces each linear part to fix a spanning set of ${\mathbb{R}}^n$.