# Difference between revisions of "Cartan-Hadamard theorem"

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Here are some equivalent formulations: | Here are some equivalent formulations: | ||

− | + | # Any [[complete Riemannian manifold|complete]] [[nonpositively curved manifold]], viz., any manifold which has nonpositive sectional curvature everywhere, has the property that its universal cover is diffeomorphic to real Euclidean space, In fact, the [[exponential map at a point|exponential map]] at any point is a covering map. | |

− | + | # Any complete simply connected nonpositively curved manifold is diffeomorphic to <math>\R^n</math> (such a manifold is termed a [[CH-manifold]]). In fact, the exponential map at any point is a diffeomorphism. | |

The equivalence follows from the fact that the universal cover of a Riemannian manifold can be given the pullback metric, in which case the range of values taken by the sectional curvature is the same for both spaces. | The equivalence follows from the fact that the universal cover of a Riemannian manifold can be given the pullback metric, in which case the range of values taken by the sectional curvature is the same for both spaces. | ||

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The Bonnet-Myers theorem states that the universal cover of a [[complete Riemannian manifold]] with [[Ricci curvature]] bounded below by a positive number, is compact. | The Bonnet-Myers theorem states that the universal cover of a [[complete Riemannian manifold]] with [[Ricci curvature]] bounded below by a positive number, is compact. | ||

− | == | + | ==Facts used== |

− | + | # [[Hopf-Rinow theorem]] | |

− | + | # [[Nonpositively curved implies conjugate-free]] | |

− | + | # [[Local isometry of complete Riemannian manifolds is covering map]] | |

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## Latest revision as of 13:12, 22 May 2008

*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*

*This article makes a prediction about the universal cover of a manifold based on given data at the level of a:*Riemannian manifold

*This result is valid in all dimensions*

## Statement

Here are some equivalent formulations:

- Any complete nonpositively curved manifold, viz., any manifold which has nonpositive sectional curvature everywhere, has the property that its universal cover is diffeomorphic to real Euclidean space, In fact, the exponential map at any point is a covering map.
- Any complete simply connected nonpositively curved manifold is diffeomorphic to (such a manifold is termed a CH-manifold). In fact, the exponential map at any point is a diffeomorphism.

The equivalence follows from the fact that the universal cover of a Riemannian manifold can be given the pullback metric, in which case the range of values taken by the sectional curvature is the same for both spaces.

## Relation with other results

### Bonnet-Myers theorem

`Further information: Bonnet-Myers theorem`

The Bonnet-Myers theorem states that the universal cover of a complete Riemannian manifold with Ricci curvature bounded below by a positive number, is compact.