Bieberbach theorem

This article describes a result related to the sectional curvature of a Riemannian manifold

This result relates information on curvature to information on topology of a manifold

Statement

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).