Myers-Bonnet theorem - Revision history

Vipul: 9 revisions

2008-05-18T19:50:02Z

9 revisions

← Older revision	Revision as of 19:50, 18 May 2008
(No difference)

Vipul at 10:50, 23 June 2007

2007-06-23T10:50:33Z

← Older revision		Revision as of 10:50, 23 June 2007
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	~~{{sectional curvature result}}~~		#redirect [[Bonnet-Myers theorem]]

	~~{{relating curvature to topology}}~~

	~~{{universal cover prediction\|topological manifold}}~~

	~~{{in all dimensions}}~~

	~~==Statement==~~

	~~If a~~ [[~~Riemannian manifold]] has the property that there exists a positive constant that lower~~-bounds the sectional curvature for all tangent planes 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.

	~~==Relation with other results==~~

	~~===Cartan-Hadamard theorem===~~

	~~{{further\|[[Cartan-Hadamard~~ theorem]]}}
	The Cartan-Hadamard theorem talks of the analogous statement when the manifold has negative curvature throughout. It says that under that assumption, the universal cover is diffeomorphic to real Euclidean space.

	Together, the Cartan-Hadamard theorem and Myers-Bott theorem tell us that a manifold which has positive curvature bounded from below, cannot be diffeomorphic to a manifold which has negative sectional curvature throughout.

Vipul: Myers-Bott theorem moved to Myers-Bonnet theorem: Name correction

2007-06-18T14:33:17Z

Myers-Bott theorem moved to Myers-Bonnet theorem: Name correction

← Older revision	Revision as of 14:33, 18 June 2007
(No difference)

Vipul: /* Cartan-Hadamard theorem */

2007-06-18T13:10:48Z

Cartan-Hadamard theorem

← Older revision		Revision as of 13:10, 18 June 2007
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	===Cartan-Hadamard theorem===		===Cartan-Hadamard theorem===

	{{further\|Cartan-Hadamard theorem]]}}		{{further\|[[Cartan-Hadamard theorem]]}}
	The Cartan-Hadamard theorem talks of the analogous statement when the manifold has negative curvature throughout. It says that under that assumption, the universal cover is diffeomorphic to real Euclidean space.		The Cartan-Hadamard theorem talks of the analogous statement when the manifold has negative curvature throughout. It says that under that assumption, the universal cover is diffeomorphic to real Euclidean space.

	Together, the Cartan-Hadamard theorem and Myers-Bott theorem tell us that a manifold which has positive curvature bounded from below, cannot be diffeomorphic to a manifold which has negative sectional curvature throughout.		Together, the Cartan-Hadamard theorem and Myers-Bott theorem tell us that a manifold which has positive curvature bounded from below, cannot be diffeomorphic to a manifold which has negative sectional curvature throughout.

Vipul: /* Relation with other results */

2007-06-18T13:10:18Z

Relation with other results

← Older revision		Revision as of 13:10, 18 June 2007
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	===Cartan-Hadamard theorem===		===Cartan-Hadamard theorem===

			{{further\|Cartan-Hadamard theorem]]}}
	The Cartan-Hadamard theorem talks of the analogous statement when the manifold has negative curvature throughout. It says that under that assumption, the universal cover is diffeomorphic to real Euclidean space.		The Cartan-Hadamard theorem talks of the analogous statement when the manifold has negative curvature throughout. It says that under that assumption, the universal cover is diffeomorphic to real Euclidean space.

	Together, the Cartan-Hadamard theorem and Myers-Bott theorem tell us that a manifold which has positive curvature bounded from below, cannot be diffeomorphic to a manifold which has negative sectional curvature throughout.		Together, the Cartan-Hadamard theorem and Myers-Bott theorem tell us that a manifold which has positive curvature bounded from below, cannot be diffeomorphic to a manifold which has negative sectional curvature throughout.

Vipul at 05:32, 11 May 2007

2007-05-11T05:32:42Z

← Older revision		Revision as of 05:32, 11 May 2007
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	{{universal cover prediction\|topological manifold}}		{{universal cover prediction\|topological manifold}}

			{{in all dimensions}}

	==Statement==		==Statement==

Vipul at 05:21, 11 May 2007

2007-05-11T05:21:32Z

← Older revision		Revision as of 05:21, 11 May 2007
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	{{relating curvature to topology}}		{{relating curvature to topology}}

	{{universal cover prediction}}		{{universal cover prediction\|topological manifold}}

	==Statement==		==Statement==

Vipul at 05:18, 11 May 2007

2007-05-11T05:18:01Z

← Older revision		Revision as of 05:18, 11 May 2007
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	{{sectional curvature result}}		{{sectional curvature result}}

			{{relating curvature to topology}}

			{{universal cover prediction}}

	==Statement==		==Statement==

Vipul at 11:17, 5 April 2007

2007-04-05T11:17:10Z

← Older revision		Revision as of 11:17, 5 April 2007
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	==Statement==		==Statement==

	If a [[Riemannian manifold]] has the property that there exists a positive constant that lower-bounds the sectional curvature for all tangent planes at all points, then the manifold is compact with finite fundamental group.		If a [[Riemannian manifold]] has the property that there exists a positive constant that lower-bounds the sectional curvature for all tangent planes 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.

	==Relation with other results==		==Relation with other results==

Vipul: Started the page

2007-03-08T16:59:23Z

Started the page

New page

{{sectional curvature result}}

==Statement==

If a [[Riemannian manifold]] has the property that there exists a positive constant that lower-bounds the sectional curvature for all tangent planes at all points, then the manifold is compact with finite fundamental group.

==Relation with other results==

===Cartan-Hadamard theorem===

The Cartan-Hadamard theorem talks of the analogous statement when the manifold has negative curvature throughout. It says that under that assumption, the universal cover is diffeomorphic to real Euclidean space.

Together, the Cartan-Hadamard theorem and Myers-Bott theorem tell us that a manifold which has positive curvature bounded from below, cannot be diffeomorphic to a manifold which has negative sectional curvature throughout.