# Bieberbach theorem

Let $M$ be a compact flat Riemannian manifold of dimension $n$. Then:
• $\pi_1(M)$ (the fundamental group) of $M$) contains a free Abelian normal subgroup of rank $n$ and finite index
• Thus $M$ is a finite quotient of a flat torus (using the fact that the only Riemannian manifolds whose fundamental groups are free Abelian, are the flat tori).