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).