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en>Rjwilmsi: Added 1 doi to a journal cite using AWB (10201)

2014-05-22T12:14:48Z

Added 1 doi to a journal cite using AWB (10201)

← Older revision		Revision as of 14:14, 22 May 2014
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	~~In mathematics, '''Hopf conjecture''' may refer to one of several conjectural statements from [[differential geometry]] and [[topology]] attributed~~ to ~~[[Heinz Hopf]].~~		Nice to meet you, my title is Refugia. For many years he's been living in North Dakota and his family members enjoys it. I am a meter reader. One of the things she loves most is to read comics and she'll be starting some thing else along with it.<br><br>Check out my blog [http://www.videokeren.com/user/FJWW http://www.videokeren.com/user/FJWW]

	~~== Positively curved Riemannian manifolds ==~~

	~~: ''A compact~~, ~~even-dimensional [[Riemannian manifold]] with positive [[sectional curvature]] has positive [[Euler characteristic]]~~.''

	For [[differential geometry of surfaces\|surfaces]], this follows from the [[Gauss–Bonnet theorem]]. For four-dimensional manifolds, this follows from the finiteness of the [[fundamental group]] and the [[Poincaré duality]]. The conjecture has been proved for manifolds of dimension 4''k''+2 or 4''k''+4 admitting an isometric [[torus action]] of a ''k''-dimensional torus and for manifolds ''M'' admitting an isometric action of a compact [[Lie group]] ''G'' with principal isotropy subgroup ''H'' and ~~cohomogeneity ''k'' such that~~

	~~: <math> k-(\operatorname{rank} G-\operatorname{rank} H)\leq 5~~. ~~</math>~~

	In a ~~related conjecture, "positive" is replaced with "nonnegative"~~.

	~~== Riemannian symmetric spaces ==~~
	~~: ''A compact [[symmetric space]] of rank greater than one cannot carry a Riemannian metric~~ of ~~positive sectional curvature.''~~

	~~In particular,~~ the ~~four-dimensional manifold ''S''<sup>2</sup>×''S'~~'~~<sup>2</sup> should admit no [[Riemannian metric]]~~ with ~~positive sectional curvature~~.

	~~== Aspherical manifolds ==~~
	~~: ''Suppose ''M''~~<~~sup~~>~~2''k''~~<~~/sup~~> ~~is a closed,~~ [~~[aspherical manifold\|aspherical]] manifold of even dimension. Then its Euler characteristic satisfies the inequality''~~

	:~~: <math> (-1)^k\chi(M^{2k})\geq 0. <~~/~~math>~~

	This topological version of Hopf conjecture for [[Riemannian manifold]]s is due to [[William Thurston]]. Ruth Charney and Mike Davis conjectured that the same inequality holds for a nonpositively curved piecewise Euclidean (PE) manifold.

	~~== Metrics with no conjugate points==~~

	~~: ''A Riemannian metric without conjugate points on n-dimensional torus is flat."~~

	~~Proved by D. Burago and S. Ivanov <ref>D. Burago and S~~. ~~Ivanov, Riemannian tori without conjugate points are flat, GEOMETRIC AND FUNCTIONAL ANALYSIS~~
	~~Volume 4, Number 3 (1994), 259-269, DOI: 10~~.~~1007/BF01896241<~~/~~ref>~~

	~~== References ==~~
	~~<references~~/>
	* Thomas Püttmann and Catherine Searle, [http://www.~~ams~~.~~org/proc/2002-130-01~~/~~S0002-9939-01-06039-7~~/~~S0002-9939-01-06039-7.pdf ''The Hopf conjecture for manifolds with low cohomogeneity or high symmetry rank''], Proc AMS, 130:1 (2001), pp 163–166~~

	~~[[Category:Differential geometry]]~~
	~~[[Category:Topology]]~~
	~~[[Category:Conjectures]~~]

184.58.2.251: /* See also */

2013-11-04T00:03:24Z

See also

← Older revision		Revision as of 02:03, 4 November 2013
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	~~Hello~~. ~~Allow me introduce [http~~:~~//simple~~-~~crafts~~.~~com/groups/best~~-~~techniques~~-for-~~keeping~~-~~yeast~~-~~infections-under-control~~/ ~~over the counter std test~~] ~~author~~. ~~Her name is Refugia Shryock. He utilized to be unemployed but now he~~ is a ~~meter reader~~. ~~Her family life in Minnesota~~. ~~He is really fond~~ of ~~doing ceramics but he~~ is ~~struggling~~ to ~~find time~~ for it.		In mathematics, '''Hopf conjecture''' may refer to one of several conjectural statements from [[differential geometry]] and [[topology]] attributed to [[Heinz Hopf]].

			== Positively curved Riemannian manifolds ==

			: ''A compact, even-dimensional [[Riemannian manifold]] with positive [[sectional curvature]] has positive [[Euler characteristic]].''

			For [[differential geometry of surfaces\|surfaces]], this follows from the [[Gauss–Bonnet theorem]]. For four-dimensional manifolds, this follows from the finiteness of the [[fundamental group]] and the [[Poincaré duality]]. The conjecture has been proved for manifolds of dimension 4''k''+2 or 4''k''+4 admitting an isometric [[torus action]] of a ''k''-dimensional torus and for manifolds ''M'' admitting an isometric action of a compact [[Lie group]] ''G'' with principal isotropy subgroup ''H'' and cohomogeneity ''k'' such that

			: <math> k-(\operatorname{rank} G-\operatorname{rank} H)\leq 5. </math>

			In a related conjecture, "positive" is replaced with "nonnegative".

			== Riemannian symmetric spaces ==
			: ''A compact [[symmetric space]] of rank greater than one cannot carry a Riemannian metric of positive sectional curvature.''

			In particular, the four-dimensional manifold ''S''<sup>2</sup>×''S''<sup>2</sup> should admit no [[Riemannian metric]] with positive sectional curvature.

			== Aspherical manifolds ==
			: ''Suppose ''M''<sup>2''k''</sup> is a closed, [[aspherical manifold\|aspherical]] manifold of even dimension. Then its Euler characteristic satisfies the inequality''

			:: <math> (-1)^k\chi(M^{2k})\geq 0. </math>

			This topological version of Hopf conjecture for [[Riemannian manifold]]s is due to [[William Thurston]]. Ruth Charney and Mike Davis conjectured that the same inequality holds for a nonpositively curved piecewise Euclidean (PE) manifold.

			== Metrics with no conjugate points==

			: ''A Riemannian metric without conjugate points on n-dimensional torus is flat."

			Proved by D. Burago and S. Ivanov <ref>D. Burago and S. Ivanov, Riemannian tori without conjugate points are flat, GEOMETRIC AND FUNCTIONAL ANALYSIS
			Volume 4, Number 3 (1994), 259-269, DOI: 10.1007/BF01896241</ref>

			== References ==
			<references/>
			* Thomas Püttmann and Catherine Searle, [http://www.ams.org/proc/2002-130-01/S0002-9939-01-06039-7/S0002-9939-01-06039-7.pdf ''The Hopf conjecture for manifolds with low cohomogeneity or high symmetry rank''], Proc AMS, 130:1 (2001), pp 163–166

			[[Category:Differential geometry]]
			[[Category:Topology]]
			[[Category:Conjectures]]

en>Helpful Pixie Bot: ISBNs (Build KC)

2012-05-06T20:03:13Z

ISBNs (Build KC)

New page

Hello. Allow me introduce [http://simple-crafts.com/groups/best-techniques-for-keeping-yeast-infections-under-control/ over the counter std test] author. Her name is Refugia Shryock. He utilized to be unemployed but now he is a meter reader. Her family life in Minnesota. He is really fond of doing ceramics but he is struggling to find time for it.