Difference between revisions of "Bonnet-Myers theorem"
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| − | + | {{Ricci curvature result}} | |
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| + | {{sectional curvature result}} | ||
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| + | {{relating curvature to topology}} | ||
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| + | {{universal cover prediction|topological manifold}} | ||
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| + | {{in all dimensions}} | ||
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| + | ==Statement== | ||
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| + | ===Ricci curvature version=== | ||
| + | |||
| + | Here are some equivalent forms: | ||
| + | * Any [[complete Riemannian manifold|complete]] simply connected Riemannian manifold whose Ricci curvature everywhere is bounded below by a positive constant, must be [[compact manifold|compact]] | ||
| + | * Any [[complete Riemannian manifold]] whose Ricci curvature everywhere is bounded below by a positive constant, must be [[compact manifold|compact]] | ||
| + | * If a [[Riemannian manifold]] has the property that there exists a positive constant that lower-bounds the | ||
| + | Ricci curvature for all tangent lines at all points, then the manifold is compact with finite fundamental group. This is equivalent to saying that the universal cover of the manifold is compact. | ||
| + | |||
| + | The reason why all these formulations are equivalent is that given any Riemannian manifold, we can consider its universal cover and give the universal cover the pullback metric. In that case, the range of values taken by the Ricci curvature is same for both manifolds. | ||
| + | |||
| + | ===Sectional curvature version=== | ||
| + | |||
| + | This makes a stronger assumption, but also gives a stronger result, namely: | ||
| + | |||
| + | Any [[complete Riemannian manifold]] whose [[sectional curvature]] is everywhere bounded below by a constant <math>k</math>, satisfies the property that its diameter (viz, maximum possible distance between two points) is at most <math>\pi/\sqrt{k}</math>. | ||
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| + | Note that an upper bound on the distance between points implies, for a complete Riemannian manifold, that it is compact. | ||
| + | ==Facts== | ||
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| + | Any [[Einstein metric]] with positive cosmological constant, for instance, satisfies the hypotheses for the Bonnet-Myers theorem, and hence, any manifold possessing such a metric is compact with finite fundamental group. | ||
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| + | ==Relation with other results== | ||
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| + | ===Cartan-Hadamard theorem=== | ||
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| + | {{further|[[Cartan-Hadamard theorem]]}} | ||
| + | The Cartan-Hadamard theorem talks of the analogous statement when the manifold has nonpositive [[sectional curvature]] throughout. It says that under that assumption, the universal cover is diffeomorphic to real Euclidean space. | ||
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| + | Together, the Cartan-Hadamard theorem and Bonnet-Myers theorem tell us that a complete manifold which has positive curvature bounded from below, cannot be diffeomorphic to a manifold which has negative sectional curvature throughout. | ||
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| + | ==Proof== | ||
| + | |||
| + | ===Proof of the sectional curvature version=== | ||
| + | |||
| + | The proof hinges on the following result: | ||
| + | |||
| + | If the sectional curvature is bounded below by a positive constant <math>k</math>, then any [[normal geodesic]] <math>c:[0,a] \to M</math> of length at least <math>\pi/\sqrt{k}</math> contains a conjugate point to its starting point. | ||
| + | |||
| + | Completeness tells us that any two points are joined by a minimal geodesic that does not contain any pair of conjugate points other than possibly the endpoints (by the [[Hopf-Rinow theorem]]). Hence the distance between any two points cannot be more than <math>\pi/\sqrt{k}</math>. | ||
Latest revision as of 19:33, 18 May 2008
This article describes a result related to the Ricci curvature of a Riemannian manifold
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:topological manifold
This result is valid in all dimensions
Contents
Statement
Ricci curvature version
Here are some equivalent forms:
- Any complete simply connected Riemannian manifold whose Ricci curvature everywhere is bounded below by a positive constant, must be compact
- Any complete Riemannian manifold whose Ricci curvature everywhere is bounded below by a positive constant, must be compact
- If a Riemannian manifold has the property that there exists a positive constant that lower-bounds the
Ricci curvature for all tangent lines at all points, then the manifold is compact with finite fundamental group. This is equivalent to saying that the universal cover of the manifold is compact.
The reason why all these formulations are equivalent is that given any Riemannian manifold, we can consider its universal cover and give the universal cover the pullback metric. In that case, the range of values taken by the Ricci curvature is same for both manifolds.
Sectional curvature version
This makes a stronger assumption, but also gives a stronger result, namely:
Any complete Riemannian manifold whose sectional curvature is everywhere bounded below by a constant 
, satisfies the property that its diameter (viz, maximum possible distance between two points) is at most 
.
Note that an upper bound on the distance between points implies, for a complete Riemannian manifold, that it is compact.
Facts
Any Einstein metric with positive cosmological constant, for instance, satisfies the hypotheses for the Bonnet-Myers theorem, and hence, any manifold possessing such a metric is compact with finite fundamental group.
Relation with other results
Cartan-Hadamard theorem
Further information: Cartan-Hadamard theorem The Cartan-Hadamard theorem talks of the analogous statement when the manifold has nonpositive sectional curvature throughout. It says that under that assumption, the universal cover is diffeomorphic to real Euclidean space.
Together, the Cartan-Hadamard theorem and Bonnet-Myers theorem tell us that a complete manifold which has positive curvature bounded from below, cannot be diffeomorphic to a manifold which has negative sectional curvature throughout.
Proof
Proof of the sectional curvature version
The proof hinges on the following result:
If the sectional curvature is bounded below by a positive constant , then any normal geodesic ![c:[0,a] \to M](/w/images/math/2/b/e/2bea427f72b3889bb89c49a420b22116.png)
of length at least contains a conjugate point to its starting point.
Completeness tells us that any two points are joined by a minimal geodesic that does not contain any pair of conjugate points other than possibly the endpoints (by the Hopf-Rinow theorem). Hence the distance between any two points cannot be more than .
