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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| - Sphärensatz (de)
- Théorème de la sphère (fr)
- Sphere theorem (en)
- Теорема о сфере (дифференциальная геометрия) (ru)
- Теорема про сферу (диференціальна геометрія) (uk)
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| - Der Sphärensatz ist ein bedeutendes Resultat aus der globalen riemannschen Geometrie. Nach Vorarbeiten von Harry Rauch bewiesen Wilhelm Klingenberg und Marcel Berger diesen Satz im Jahr 1961. (de)
- En géométrie riemannienne, le théorème de la sphère montre que des informations sur la courbure d'une variété, sorte d'espace courbe à plusieurs dimensions, peuvent contraindre fortement la topologie, c'est-à-dire la forme globale de cet espace. Le théorème original est établi en 1960-61 par Marcel Berger et Wilhelm Klingenberg, comme généralisation d'un premier résultat de (en) de 1951. Il a été considérablement amélioré en 2007 par Simon Brendle et Richard Schoen. Cette nouvelle version du théorème est parfois appelée théorème de la sphère différentiable. (fr)
- Теорема о сфере — общее название теорем, дающих достаточные условия на риманову метрику, гарантирующие гомеоморфность или диффеоморфность многообразия стандартной сфере. (ru)
- Теорема про сферу — загальна назва теорем, що дають достатні умови на ріманову метрику, які гарантують гомеоморфність або дифеоморфність многовиду стандартній сфері. (uk)
- 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. (en)
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| - Der Sphärensatz ist ein bedeutendes Resultat aus der globalen riemannschen Geometrie. Nach Vorarbeiten von Harry Rauch bewiesen Wilhelm Klingenberg und Marcel Berger diesen Satz im Jahr 1961. (de)
- En géométrie riemannienne, le théorème de la sphère montre que des informations sur la courbure d'une variété, sorte d'espace courbe à plusieurs dimensions, peuvent contraindre fortement la topologie, c'est-à-dire la forme globale de cet espace. Le théorème original est établi en 1960-61 par Marcel Berger et Wilhelm Klingenberg, comme généralisation d'un premier résultat de (en) de 1951. Il a été considérablement amélioré en 2007 par Simon Brendle et Richard Schoen. Cette nouvelle version du théorème est parfois appelée théorème de la sphère différentiable. (fr)
- 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. Note that the conclusion is false if the sectional curvatures are allowed to take values in the closed interval . 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 symmetric spaces. (en)
- Теорема о сфере — общее название теорем, дающих достаточные условия на риманову метрику, гарантирующие гомеоморфность или диффеоморфность многообразия стандартной сфере. (ru)
- Теорема про сферу — загальна назва теорем, що дають достатні умови на ріманову метрику, які гарантують гомеоморфність або дифеоморфність многовиду стандартній сфері. (uk)
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