Conjugate-free Riemannian manifold - Revision history

Vipul: 5 revisions

2008-05-18T19:34:56Z

5 revisions

← Older revision	Revision as of 19:34, 18 May 2008
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Vipul: /* Stronger properties */

2007-07-12T06:26:14Z

Stronger properties

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	===Stronger properties===		===Stronger properties===

	* [[Nonpositively curved manifold]]		* [[Nonpositively curved manifold]]: {{proofat\|[[Nonpositively curved implies conjugate-free]]}}

	===Weaker properties===		===Weaker properties===

	* [[Nonpositively curved manifold]] if we assume compactness		* [[Nonpositively curved manifold]] if we assume compactness

Vipul at 06:25, 12 July 2007

2007-07-12T06:25:30Z

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	===Stronger properties===		===Stronger properties===

	* [[~~Negatively~~ curved manifold]]		* [[Nonpositively curved manifold]]

	===Weaker properties===		===Weaker properties===

	* [[Nonpositively curved manifold]] if we assume compactness		* [[Nonpositively curved manifold]] if we assume compactness

Vipul at 06:20, 12 July 2007

2007-07-12T06:20:57Z

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	{{Riemannian ~~manifold~~ property}}		{{Riemannian metric property}}

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Vipul: /* Weaker properties */

2007-07-06T15:29:10Z

Weaker properties

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	===Weaker properties===		===Weaker properties===

	* [[Nonpositively curved manifold]]		* [[Nonpositively curved manifold]] if we assume compactness

Vipul at 15:14, 6 July 2007

2007-07-06T15:14:23Z

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{{Riemannian manifold property}}

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==History==

The study of conjugate-free Riemannian manifolds (or Riemannian manifolds without conjugate points) originated with a theorem by [[Hopf]] that the only Riemannian metric on the torus without conjugate points is the flat one. Green generalized this to say that any Riemannian metric without conjugate points, must have everywhere nonpositive sectional curvature.

==Definition==

A Riemannian manifold is said to be '''conjugate-free''' or '''without conjugate points''' if it does not contain any pair of conjugate points. In other words, there is no pair of points for which there is a smoothly varying family of geodesics joining them.

==Relation with other properties==

===Stronger properties===

* [[Negatively curved manifold]]

===Weaker properties===

* [[Nonpositively curved manifold]]