Wien approximation - Revision history

en>Headbomb: /* top */ overlinking

2014-08-09T15:11:36Z

top: overlinking

← Older revision		Revision as of 17:11, 9 August 2014
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	~~{{about\|sphere theorem for Riemannian manifolds\|a different result by~~ the ~~same~~ name~~\|Sphere theorem (3-manifolds)}}~~		Hello. Allow me introduce the author. Her name is Refugia Shryock. His spouse doesn't like it the way he does but what he really likes performing is to do aerobics and he's been performing it for fairly a whilst. Years in the past he moved to North Dakota and std home test his [http://www.revleft.com/vb/member.php?u=160656 at home std test] family members loves it. [http://welkinn.com/profile-17521/info/ std testing at home] Hiring has been my [http://ohnotheydidnt.Livejournal.com/49721712.html profession] for some time but I've currently applied for an additional 1.<br><br>my blog :: [http://www.crmidol.com/discussion/19701/how-tell-if-woman-interested-or-sexually-captivated-you home std test kit]

	In [[Riemannian geometry]], the '''sphere theorem''', also known as the '''quarter-pinched sphere theorem''', strongly restricts the topology of manifolds admitting metrics with a particular curvature bound. The precise statement of the theorem is ~~as follows~~. If '~~'M'' is a [[Complete metric\|complete]], [[simply-connected]], ''n''-dimensional [[Riemannian manifold]] with [[sectional curvature]] taking values in~~ the ~~interval <math>(1,4]</math> then ''M''~~ is ~~[[homeomorphic]]~~ to ~~the [[n-sphere\|~~''n''-sphere]]. (To be precise, we mean the sectional curvature of every tangent 2-plane at each point must lie in <math>(1,4]</math>.) Another way of stating the result is that if ''M'' is not homeomorphic to the sphere, then it ~~is impossible to put~~ a ~~metric on ''M'' with quarter-pinched curvature~~.

	~~Note that the conclusion is false if~~ the ~~sectional curvatures are allowed~~ to take values in the ''closed'' interval <math>[1,4]</math>. The standard counterexample is [[complex projective space]] with the [[Fubini-Study metric]]; sectional curvatures of this metric take on values between 1 and ~~4, with endpoints included.~~ ~~Other counterexamples may be found among the rank one [[Riemannian symmetric space\|symmetric spaces]].~~

	~~==Differentiable sphere theorem==~~
	~~The original proof of the sphere theorem did not conclude that ''M'' was necessarily [[diffeomorphic]] to the ''n''-sphere.~~ ~~This complication is because spheres in higher dimensions admit~~ [~~[smooth structure]]s that are not diffeomorphic~~. ~~(For more information, see the article on [[exotic sphere]]s~~.) However, in 2007 [[Simon Brendle]] and [[Richard Schoen]] utilized [[Ricci flow]] to prove that with the above hypotheses, ''M'' is necessarily diffeomorphic to the ''n''-sphere with its standard smooth structure. Moreover, the proof of Brendle and Schoen only uses the weaker assumption of pointwise rather than global pinching. ~~This result is known as the '''Differentiable Sphere Theorem'''.~~

	=~~=History of the sphere theorem==~~
	~~[[Heinz Hopf\|Hopf~~]~~] conjectured that a simply connected manifold with pinched sectional curvature is a sphere~~. ~~In 1951,~~ [[Harry Rauch]] showed that a simply connected manifold with curvature in [3/4,1] is homeomorphic to a sphere. In 1960, Berger and Klingenberg proved the topological version of the sphere theorem with the optimal pinching constant.

	~~==References==~~
	*{{cite book \| author=Brendle, Simon \| title =Ricci Flow and the Sphere Theorem\| publisher=American Mathematical Society \| year=2010\| isbn=0-8218-4938-7 \| url=http://~~www~~.~~ams.org~~/~~bookstore~~-~~getitem~~/~~item=GSM-111}}.~~
	*{{Citation \| last1=Brendle \| first1=Simon \| last2=Schoen \| first2=Richard \| title=Manifolds with 1/~~4-pinched curvature are space forms \| id={{MathSciNet \| id = 2449060}} \| year=2009 \| journal=[~~[~~Journal of the American Mathematical Society]] \| volume=22 \| pages=287–307\|url=~~http://~~www~~.~~ams~~.~~org/jams/2009-22-01/S0894-0347-08-00613-9~~/~~S0894-0347-08-00613-9~~.~~pdf}}~~.
	*{{Citation \| last1=Brendle \| first1=Simon \| last2=Schoen \| first2=Richard \| title=Curvature, Sphere Theorems, and the Ricci Flow \| year=2011 \| journal=[[~~Bulletin of the American Mathematical Society]] \| volume=48 \| pages=1–32\|url=~~http://www.~~ams~~.~~org~~/~~journals~~/~~bull~~/~~2011~~-48-~~01/S0273-0979-2010~~-~~01312~~-~~4/S0273~~-~~0979~~-~~2010~~-~~01312~~-~~4.pdf}}.~~

	~~[[Category:Riemannian geometry]]~~
	~~[[Category:Theorems in topology]]~~
	~~[[Category:Theorems in Riemannian geometry]~~]

en>Reinschrift at 18:05, 21 August 2013

2013-08-21T18:05:57Z

← Older revision		Revision as of 20:05, 21 August 2013
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	~~She is~~ known by the ~~title~~ of ~~Myrtle Shryock~~. ~~For~~ a ~~whilst she~~'~~s been~~ in ~~South Dakota~~. ~~What I love doing~~ is ~~taking part in baseball but I haven~~'~~t produced~~ a ~~dime~~ with it. ~~Hiring~~ is ~~his occupation.~~<br><br>~~My web~~-~~site~~ - ~~std home test~~ ([http://www.~~gaysphere~~.~~net~~/~~blog~~/~~167593 visit~~ the ~~following webpage~~])		{{about\|sphere theorem for Riemannian manifolds\|a different result by the same name\|Sphere theorem (3-manifolds)}}

			In [[Riemannian geometry]], the '''sphere theorem''', also known as the '''quarter-pinched sphere theorem''', strongly restricts the topology of manifolds admitting metrics with a particular curvature bound. The precise statement of the theorem is as follows. If ''M'' is a [[Complete metric\|complete]], [[simply-connected]], ''n''-dimensional [[Riemannian manifold]] with [[sectional curvature]] taking values in the interval <math>(1,4]</math> then ''M'' is [[homeomorphic]] to the [[n-sphere\|''n''-sphere]]. (To be precise, we mean the sectional curvature of every tangent 2-plane at each point must lie in <math>(1,4]</math>.) Another way of stating the result is that if ''M'' is not homeomorphic to the sphere, then it is impossible to put a metric on ''M'' with quarter-pinched curvature.

			Note that the conclusion is false if the sectional curvatures are allowed to take values in the ''closed'' interval <math>[1,4]</math>. The standard counterexample is [[complex projective space]] with the [[Fubini-Study metric]]; sectional curvatures of this metric take on values between 1 and 4, with endpoints included. Other counterexamples may be found among the rank one [[Riemannian symmetric space\|symmetric spaces]].

			==Differentiable sphere theorem==
			The original proof of the sphere theorem did not conclude that ''M'' was necessarily [[diffeomorphic]] to the ''n''-sphere. This complication is because spheres in higher dimensions admit [[smooth structure]]s that are not diffeomorphic. (For more information, see the article on [[exotic sphere]]s.) However, in 2007 [[Simon Brendle]] and [[Richard Schoen]] utilized [[Ricci flow]] to prove that with the above hypotheses, ''M'' is necessarily diffeomorphic to the ''n''-sphere with its standard smooth structure. Moreover, the proof of Brendle and Schoen only uses the weaker assumption of pointwise rather than global pinching. This result is known as the '''Differentiable Sphere Theorem'''.

			==History of the sphere theorem==
			[[Heinz Hopf\|Hopf]] conjectured that a simply connected manifold with pinched sectional curvature is a sphere. In 1951, [[Harry Rauch]] showed that a simply connected manifold with curvature in [3/4,1] is homeomorphic to a sphere. In 1960, Berger and Klingenberg proved the topological version of the sphere theorem with the optimal pinching constant.

			==References==
			*{{cite book \| author=Brendle, Simon \| title =Ricci Flow and the Sphere Theorem\| publisher=American Mathematical Society \| year=2010\| isbn=0-8218-4938-7 \| url=http://www.ams.org/bookstore-getitem/item=GSM-111}}.
			*{{Citation \| last1=Brendle \| first1=Simon \| last2=Schoen \| first2=Richard \| title=Manifolds with 1/4-pinched curvature are space forms \| id={{MathSciNet \| id = 2449060}} \| year=2009 \| journal=[[Journal of the American Mathematical Society]] \| volume=22 \| pages=287–307\|url=http://www.ams.org/jams/2009-22-01/S0894-0347-08-00613-9/S0894-0347-08-00613-9.pdf}}.
			*{{Citation \| last1=Brendle \| first1=Simon \| last2=Schoen \| first2=Richard \| title=Curvature, Sphere Theorems, and the Ricci Flow \| year=2011 \| journal=[[Bulletin of the American Mathematical Society]] \| volume=48 \| pages=1–32\|url=http://www.ams.org/journals/bull/2011-48-01/S0273-0979-2010-01312-4/S0273-0979-2010-01312-4.pdf}}.

			[[Category:Riemannian geometry]]
			[[Category:Theorems in topology]]
			[[Category:Theorems in Riemannian geometry]]

en>Teply: miscategorized - not really foundational QM

2012-06-24T18:22:53Z

miscategorized - not really foundational QM

New page

She is known by the title of Myrtle Shryock. For a whilst she's been in South Dakota. What I love doing is taking part in baseball but I haven't produced a dime with it. Hiring is his occupation.<br><br>My web-site - std home test ([http://www.gaysphere.net/blog/167593 visit the following webpage])