Cartan-Hadamard theorem - Revision history

Vipul at 13:12, 22 May 2008

2008-05-22T13:12:01Z

← Older revision		Revision as of 13:12, 22 May 2008
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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		# 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]])		# 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~~==		==Facts used==

	~~===Main ingredient of proof===~~		# [[Hopf-Rinow theorem]]
			# [[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.~~		# [[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.~~

Vipul: 13 revisions

2008-05-18T19:34:02Z

13 revisions

← Older revision	Revision as of 19:34, 18 May 2008
(No difference)

Vipul: /* Proof details */

2007-08-31T11:11:38Z

Proof details

← Older revision		Revision as of 11:11, 31 August 2007
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	First, the following observations:		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 surjective		* 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".		* 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>.		* 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>.

Vipul: /* Statement */

2007-08-05T12:01:51Z

Statement

← Older revision		Revision as of 12:01, 5 August 2007
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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 ~~negative~~ sectional curvature everywhere, has the property that its universal cover is diffeomorphic to real Euclidean space		* 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
	* Any complete simply connected nonpositively curved manifold is diffeomorphic to <math>\R^n</math> (such a manifold is termed a [[CH-manifold]])		* Any complete simply connected nonpositively curved manifold is diffeomorphic to <math>\R^n</math> (such a manifold is termed a [[CH-manifold]])

Vipul: /* Proof */

2007-07-13T02:13:07Z

Proof

← Older revision		Revision as of 02:13, 13 July 2007
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	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.		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.

	~~Nonpositive curvature comes of use because it tells us that~~ the manifold ~~does not have~~ [[~~conjugate points~~]] ~~(viz~~, it is a [[conjugate-free Riemannian manifold]]).		===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 surjective
			* 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.

Vipul: /* Statement */

2007-07-13T02:07:42Z

Statement

← Older revision		Revision as of 02:07, 13 July 2007
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	Here are some equivalent formulations:		Here are some equivalent formulations:
	* Any complete [[nonpositively curved manifold]], viz any manifold which has negative sectional curvature everywhere, has the property that its universal cover is diffeomorphic to real Euclidean space		* Any [[complete Riemannian manifold\|complete]] [[nonpositively curved manifold]], viz any manifold which has negative sectional curvature everywhere, has the property that its universal cover is diffeomorphic to real Euclidean space
	* Any complete simply connected nonpositively curved manifold is diffeomorphic to <math>\R^n</math> (such a manifold is termed a [[CH-manifold]])		* Any complete simply connected nonpositively curved manifold is diffeomorphic to <math>\R^n</math> (such a manifold is termed a [[CH-manifold]])

Vipul at 06:24, 12 July 2007

2007-07-12T06:24:42Z

← Older revision		Revision as of 06:24, 12 July 2007
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	Here are some equivalent formulations:		Here are some equivalent formulations:
	* Any complete [[~~negatively~~ curved manifold]], viz any manifold which has negative sectional curvature everywhere, has the property that its universal cover is diffeomorphic to real Euclidean space		* Any complete [[nonpositively curved manifold]], viz any manifold which has negative sectional curvature everywhere, has the property that its universal cover is diffeomorphic to real Euclidean space
	* Any complete simply connected ~~negatively~~ curved manifold is diffeomorphic to <math>\R^n</math> (such a manifold is termed a [[CH-manifold]])		* Any complete simply connected nonpositively curved manifold is diffeomorphic to <math>\R^n</math> (such a manifold is termed a [[CH-manifold]])

	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==

			===Main ingredient of proof===

			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.

			Nonpositive curvature comes of use because it tells us that the manifold does not have [[conjugate points]] (viz, it is a [[conjugate-free Riemannian manifold]]).

Vipul: /* Bonnet-Myers theorem */

2007-07-12T06:18:29Z

Bonnet-Myers theorem

← Older revision		Revision as of 06:18, 12 July 2007
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	{{further\|[[Bonnet-Myers theorem]]}}		{{further\|[[Bonnet-Myers theorem]]}}

	The Bonnet-Myers theorem states that the universal cover of a 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.

Vipul at 06:18, 12 July 2007

2007-07-12T06:18:01Z

← Older revision		Revision as of 06:18, 12 July 2007
Line 10:		Line 10:

	Here are some equivalent formulations:		Here are some equivalent formulations:
	* Any [[negatively curved manifold]], viz any manifold which has negative sectional curvature everywhere, has the property that its universal cover is diffeomorphic to real Euclidean space		* Any complete [[negatively curved manifold]], viz any manifold which has negative sectional curvature everywhere, has the property that its universal cover is diffeomorphic to real Euclidean space
	* Any simply connected negatively curved manifold is diffeomorphic to <math>\R^n</math> (such a manifold is termed a [[CH-manifold]])		* Any complete simply connected negatively curved manifold is diffeomorphic to <math>\R^n</math> (such a manifold is termed a [[CH-manifold]])

	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.

Vipul at 11:00, 23 June 2007

2007-06-23T11:00:08Z

← Older revision		Revision as of 11:00, 23 June 2007
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	==Statement==		==Statement==

	Any [[negatively curved manifold]], viz any manifold which has negative sectional curvature everywhere, has the property that its universal cover is diffeomorphic to real Euclidean space.		Here are some equivalent formulations:
			* Any [[negatively curved manifold]], viz any manifold which has negative sectional curvature everywhere, has the property that its universal cover is diffeomorphic to real Euclidean space
			* Any simply connected negatively curved manifold is diffeomorphic to <math>\R^n</math> (such a manifold is termed a [[CH-manifold]])

			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==		==Relation with other results==