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
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# 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]])
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# 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.
  
==Proof==
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==Facts used==
  
===Main ingredient of proof===
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# [[Hopf-Rinow theorem]]
 
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# [[Nonpositively curved implies conjugate-free]]
The main ingredient is to show that the exponential map from the tangent space at any point, to the manifold, is a well-defined covering map.
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# [[Local isometry of complete Riemannian manifolds is covering map]]
 
 
===Proof details===
 
 
 
First, the following observations:
 
 
 
* Since the manifold is [[complete Riemannian manifold|complete]], the exponential map from the tangent space at any point, to the whole manifold, is well-defined and surjective (we here invoke the [[Hopf-Rinow theorem]])
 
* Since  the manifold is nonpositively curved, it is [[conjugate-free Riemannian manifold|conjugate-free]]. In other words, no two points in it are conjugate. In other words, given any two points, any two geodesics joining the two points are "far away".
 
* We know that the map <math>\exp_p:T_pM \to M</math> looks like a covering map at any point <math>q</math> in the manifold not conjugate to <math>p</math>.
 
 
 
Piecing together these facts, the exponential map is a covering map from <math>\R^n</math> to the manifold.
 
 
 
This proves the result.
 

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:

  1. 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.
  2. Any complete simply connected nonpositively curved manifold is diffeomorphic to \R^n (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.

Facts used

  1. Hopf-Rinow theorem
  2. Nonpositively curved implies conjugate-free
  3. Local isometry of complete Riemannian manifolds is covering map