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| {{about|sphere theorem for Riemannian manifolds|a different result by the same name|Sphere theorem (3-manifolds)}}
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| 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.
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| 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]].
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| ==Differentiable sphere theorem==
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| 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'''.
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| ==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.
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| ==References==
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| *{{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}}.
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| *{{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}}.
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| *{{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}}.
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| [[Category:Riemannian geometry]]
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| [[Category:Theorems in topology]]
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| [[Category:Theorems in Riemannian geometry]]
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