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| In mathematics, '''Hopf conjecture''' may refer to one of several conjectural statements from [[differential geometry]] and [[topology]] attributed to [[Heinz Hopf]].
| | Nice to meet you, my title is Refugia. For many years he's been living in North Dakota and his family members enjoys it. I am a meter reader. One of the things she loves most is to read comics and she'll be starting some thing else along with it.<br><br>Check out my blog [http://www.videokeren.com/user/FJWW http://www.videokeren.com/user/FJWW] |
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| == Positively curved Riemannian manifolds ==
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| : ''A compact, even-dimensional [[Riemannian manifold]] with positive [[sectional curvature]] has positive [[Euler characteristic]].''
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| For [[differential geometry of surfaces|surfaces]], this follows from the [[Gauss–Bonnet theorem]]. For four-dimensional manifolds, this follows from the finiteness of the [[fundamental group]] and the [[Poincaré duality]]. The conjecture has been proved for manifolds of dimension 4''k''+2 or 4''k''+4 admitting an isometric [[torus action]] of a ''k''-dimensional torus and for manifolds ''M'' admitting an isometric action of a compact [[Lie group]] ''G'' with principal isotropy subgroup ''H'' and cohomogeneity ''k'' such that
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| : <math> k-(\operatorname{rank} G-\operatorname{rank} H)\leq 5. </math>
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| In a related conjecture, "positive" is replaced with "nonnegative".
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| == Riemannian symmetric spaces ==
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| : ''A compact [[symmetric space]] of rank greater than one cannot carry a Riemannian metric of positive sectional curvature.''
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| In particular, the four-dimensional manifold ''S''<sup>2</sup>×''S''<sup>2</sup> should admit no [[Riemannian metric]] with positive sectional curvature.
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| == Aspherical manifolds ==
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| : ''Suppose ''M''<sup>2''k''</sup> is a closed, [[aspherical manifold|aspherical]] manifold of even dimension. Then its Euler characteristic satisfies the inequality''
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| :: <math> (-1)^k\chi(M^{2k})\geq 0. </math> | |
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| This topological version of Hopf conjecture for [[Riemannian manifold]]s is due to [[William Thurston]]. Ruth Charney and Mike Davis conjectured that the same inequality holds for a nonpositively curved piecewise Euclidean (PE) manifold.
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| == Metrics with no conjugate points==
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| : ''A Riemannian metric without conjugate points on n-dimensional torus is flat."
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| Proved by D. Burago and S. Ivanov <ref>D. Burago and S. Ivanov, Riemannian tori without conjugate points are flat, GEOMETRIC AND FUNCTIONAL ANALYSIS
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| Volume 4, Number 3 (1994), 259-269, DOI: 10.1007/BF01896241</ref>
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| == References ==
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| <references/>
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| * Thomas Püttmann and Catherine Searle, [http://www.ams.org/proc/2002-130-01/S0002-9939-01-06039-7/S0002-9939-01-06039-7.pdf ''The Hopf conjecture for manifolds with low cohomogeneity or high symmetry rank''], Proc AMS, 130:1 (2001), pp 163–166
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| [[Category:Differential geometry]]
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| [[Category:Topology]]
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| [[Category:Conjectures]]
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Nice to meet you, my title is Refugia. For many years he's been living in North Dakota and his family members enjoys it. I am a meter reader. One of the things she loves most is to read comics and she'll be starting some thing else along with it.
Check out my blog http://www.videokeren.com/user/FJWW