Concept information
Preferred term
sphere theorem
Definition(s)
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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, simply-connected, n-dimensional Riemannian manifold with sectional curvature taking values in the interval (1,4] 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 (1,4].) 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.
(Wikipedia, The Free Encyclopedia, https://en.wikipedia.org/wiki/Sphere_theorem)
Broader concept(s)
Synonym(s)
- quarter-pinched sphere theorem
In other languages
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French
URI
http://data.loterre.fr/ark:/67375/PSR-MZ0GN0F0-S
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