# Bieberbach theorem

From Diffgeom

*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 be a compact flat Riemannian manifold of dimension . Then:

- (the fundamental group) of ) contains a free Abelian normal subgroup of rank and finite index
- Thus 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).