Bieberbach theorem: Difference between revisions

Revision as of 11:19, 23 June 2007

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

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).

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 Let <math>M</math> be a [[compact manifold|compact]] [[flat metric|flat]] [[Riemannian manifold]] of dimension <math>n</math>. Then:
-* <math>\pi_1(M)</math> (the [[fundamental group]]) of <math>M</math>)
+* <math>\pi_1(M)</math> (the [[fundamental group]]) of <math>M</math>) contains a free [[Abelian group|Abelian]] [[normal subgroup]] of rank <math>n</math> and finite index
+* Thus <math>M</math> 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).