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- Sphere_theorem abstract "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.".
- Sphere_theorem wikiPageExternalLink item=GSM-111.
- Sphere_theorem wikiPageExternalLink S0894-0347-08-00613-9.pdf.
- Sphere_theorem wikiPageExternalLink S0273-0979-2010-01312-4.pdf.
- Sphere_theorem wikiPageID "4685719".
- Sphere_theorem wikiPageRevisionID "599464183".
- Sphere_theorem hasPhotoCollection Sphere_theorem.
- Sphere_theorem subject Category:Riemannian_geometry.
- Sphere_theorem subject Category:Theorems_in_Riemannian_geometry.
- Sphere_theorem subject Category:Theorems_in_topology.
- Sphere_theorem type Abstraction100002137.
- Sphere_theorem type Communication100033020.
- Sphere_theorem type Message106598915.
- Sphere_theorem type Proposition106750804.
- Sphere_theorem type Statement106722453.
- Sphere_theorem type Theorem106752293.
- Sphere_theorem type TheoremsInGeometry.
- Sphere_theorem type TheoremsInRiemannianGeometry.
- Sphere_theorem type TheoremsInTopology.
- Sphere_theorem comment "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.".
- Sphere_theorem label "Sphere theorem".
- Sphere_theorem label "Sphärensatz".
- Sphere_theorem sameAs Sphärensatz.
- Sphere_theorem sameAs m.03m3q82.
- Sphere_theorem sameAs Q2309682.
- Sphere_theorem sameAs Q2309682.
- Sphere_theorem sameAs Sphere_theorem.
- Sphere_theorem wasDerivedFrom Sphere_theorem?oldid=599464183.
- Sphere_theorem isPrimaryTopicOf Sphere_theorem.