Bieberbach theorem

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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).