Bieberbach theorem
= Bieberbach theorem
{c}
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 $\mathbb R^n$.