Bonnet-Myers theorem - Revision history

Vipul: 8 revisions

2008-05-18T19:33:44Z

8 revisions

← Older revision	Revision as of 19:33, 18 May 2008
(No difference)

59.92.146.9: /* Proof of the sectional curvature version */

2007-08-30T10:42:47Z

Proof of the sectional curvature version

← Older revision		Revision as of 10:42, 30 August 2007
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	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.		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 (by the [[Hopf-Rinow theorem]]) ~~and hence~~ the distance between any two points cannot be more than <math>\pi/\sqrt{k}</math>.		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>.

Vipul: /* Cartan-Hadamard theorem */

2007-08-05T12:14:30Z

Cartan-Hadamard theorem

← Older revision		Revision as of 12:14, 5 August 2007
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	{{further\|[[Cartan-Hadamard theorem]]}}		{{further\|[[Cartan-Hadamard theorem]]}}
	The Cartan-Hadamard theorem talks of the analogous statement when the manifold has ~~negative~~ [[sectional 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 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 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 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==

Vipul at 06:41, 12 July 2007

2007-07-12T06:41:41Z

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 {{Ricci curvature result}}
 {{relating curvature to topology}}
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 ==Statement==
 Here are some equivalent forms:
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 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.
 ==Facts==
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 Together, the Cartan-Hadamard theorem and Bonnet-Myers 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: /* Statement */

2007-07-12T06:19:02Z

Statement

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	Here are some equivalent forms:		Here are some equivalent forms:
	* Any simply connected Riemannian manifold whose Ricci curvature everywhere is bounded below by a positive constant, must be [[compact manifold\|compact]]		* 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 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		* 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.		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.

Vipul at 11:06, 23 June 2007

2007-06-23T11:06:00Z

← Older revision		Revision as of 11:06, 23 June 2007
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	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.		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.

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

Vipul at 10:56, 23 June 2007

2007-06-23T10:56:05Z

← Older revision		Revision as of 10:56, 23 June 2007
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	==Statement==		==Statement==

	If a [[Riemannian manifold]] has the property that there exists a positive constant that lower-bounds the		Here are some equivalent forms:
			* Any simply connected Riemannian manifold whose Ricci curvature everywhere is bounded below by a positive constant, must be [[compact manifold\|compact]]
			* Any 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.		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.

	~~Note~~ that the ~~condition on~~ Ricci curvature is ~~weaker than the corresponding condition on sectional curvature~~.		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.

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

Vipul at 10:50, 23 June 2007

2007-06-23T10:50:20Z

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	~~#redirect~~ [[~~Myers~~-~~Bonnet~~ theorem]]		{{Ricci curvature result}}

			{{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
			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.

			Note that the condition on Ricci curvature is weaker than the corresponding condition on sectional curvature.

			==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 [[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 manifold which has positive curvature bounded from below, cannot be diffeomorphic to a manifold which has negative sectional curvature throughout.

Vipul at 14:44, 18 June 2007

2007-06-18T14:44:15Z

New page

#redirect [[Myers-Bonnet theorem]]