Cheeger-Gromoll conjecture - Revision history

Vipul: 3 revisions

2008-05-18T19:34:24Z

3 revisions

← Older revision	Revision as of 19:34, 18 May 2008
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Vipul: /* Gromoll-Meyer theorem */

2007-08-30T10:45:08Z

Gromoll-Meyer theorem

← Older revision		Revision as of 10:45, 30 August 2007
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	===Gromoll-Meyer theorem===		===Gromoll-Meyer theorem===

	This proves a weaker form of the conjecture where ''positivity'' ~~is replaced by~~ ''quasi-positivity''.		This proves a weaker form of the conjecture where ''positivity'' replaces ''quasi-positivity''.

	==References==		==References==

	* ''On the structure of complete manifolds of nonnegative curvature'' by Jeff Cheeger and Detlef Gromoll		* ''On the structure of complete manifolds of nonnegative curvature'' by Jeff Cheeger and Detlef Gromoll

Vipul at 15:41, 7 July 2007

2007-07-07T15:41:33Z

← Older revision		Revision as of 15:41, 7 July 2007
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	{{~~conjecture~~}}		{{sectional curvature result}}

			==History==

			===Proposal as a conjecture===

			The conjecture was made by Cheeger and Gromoll in their celebrated paper ''On the structure of complete manifolds of nonnegative curvature''.

			===Proof===

			The conjecture was proved by work of Perelman, following from [[Perelman rigidity theorem]].

	==Statement==		==Statement==
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	Any [[complete Riemannian manifold\|complete]] [[open Riemannian manifold\|open]] [[quasi-positively curved Riemannian manifold]] is diffeomorphic to <math>\R^n</math>.		Any [[complete Riemannian manifold\|complete]] [[open Riemannian manifold\|open]] [[quasi-positively curved Riemannian manifold]] is diffeomorphic to <math>\R^n</math>.

	==~~Progress towards the conjecture~~==		==Relation with other results==

	===Cohn-Vossen theorem===		===Cohn-Vossen theorem===

Vipul at 15:17, 7 July 2007

2007-07-07T15:17:19Z

New page

{{conjecture}}

==Statement==

Any [[complete Riemannian manifold|complete]] [[open Riemannian manifold|open]] [[quasi-positively curved Riemannian manifold]] is diffeomorphic to <math>\R^n</math>.

==Progress towards the conjecture==

===Cohn-Vossen theorem===

The conjecture is known to be true in dimension two. This is the content of the [[Cohn-Vossen theorem]].

===Gromoll-Meyer theorem===

This proves a weaker form of the conjecture where ''positivity'' is replaced by ''quasi-positivity''.

==References==

* ''On the structure of complete manifolds of nonnegative curvature'' by Jeff Cheeger and Detlef Gromoll