Sphere theorem
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, simply-connected, n-dimensional Riemannian manifold with sectional curvature taking values in the interval then M is homeomorphic to the n-sphere. (To be precise, we mean the sectional curvature of every tangent 2-plane at each point must lie in .) 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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Classification of manifoldsComplex projective planeDifferentiable sphere theoremFrankel conjectureFubini–Study metricHarry RauchHopf conjectureList of differential geometry topicsList of geometric topology topicsList of theoremsMikhael Gromov (mathematician)Quarter-pinched sphere theoremRicci flowRichard S. HamiltonRichard SchoenRiemannian geometrySchur's lemma (Riemannian geometry)Simon BrendleWilhelm Klingenberg
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Sphere theorem
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, simply-connected, n-dimensional Riemannian manifold with sectional curvature taking values in the interval then M is homeomorphic to the n-sphere. (To be precise, we mean the sectional curvature of every tangent 2-plane at each point must lie in .) 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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Der Sphärensatz ist ein bedeut ...... rger diesen Satz im Jahr 1961.
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En géométrie riemannienne, le ...... e de la sphère différentiable.
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In Riemannian geometry, the sp ...... the rank one symmetric spaces.
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Теорема о сфере — общее назван ...... ногообразия стандартной сфере.
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Der Sphärensatz ist ein bedeut ...... rger diesen Satz im Jahr 1961.
@de
En géométrie riemannienne, le ...... e de la sphère différentiable.
@fr
In Riemannian geometry, the sp ...... ith quarter-pinched curvature.
@en
Теорема о сфере — общее назван ...... ногообразия стандартной сфере.
@ru
label
Sphere theorem
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Sphärensatz
@de
Théorème de la sphère
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Теорема о сфере (дифференциальная геометрия)
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