Bayesian average: Difference between revisions

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In [[complex geometry]], a '''Hopf manifold''' {{harv|Hopf|1948}}  is obtained
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as a quotient of the complex [[vector space]]
(with zero deleted) <math>({\Bbb C}^n\backslash 0)</math>
by a [[Group action|free action]] of the [[Group (mathematics)|group]] <math>\Gamma \cong {\Bbb Z}</math> of
[[integer]]s, with the generator <math>\gamma</math>
of <math>\Gamma</math> acting by holomorphic [[Contraction mapping|contractions]]. Here, a ''holomorphic contraction''
is a map <math>\gamma:\; {\Bbb C}^n \mapsto  {\Bbb C}^n</math>
such that a sufficiently big iteration <math>\;\gamma^N</math>
puts any given [[compact subset]] <math>{\Bbb C}^n</math>
onto an arbitrarily small [[Neighbourhood (mathematics)|neighbourhood]] of 0.
 
Two dimensional Hopf manifolds are called [[Hopf surface]]s.
 
== Examples ==
In a typical situation, <math>\Gamma</math> is generated
by a linear contraction, usually a [[diagonal matrix]]
<math>q\cdot Id</math>, with <math>q\in {\Bbb C}</math>
a complex number, <math>0<|q|<1</math>. Such manifold
is called ''a classical Hopf manifold''.
 
== Properties ==
A Hopf manifold <math>H:=({\Bbb C}^n\backslash 0)/{\Bbb Z}</math>  
is [[diffeomorphic]] to <math>S^{2n-1}\times S^1</math>.
For <math>n\geq 2</math>, it is non-[[Kähler manifold|Kähler]]. Indeed,
the first [[cohomology group]] of ''H''
is odd-dimensional. By [[Hodge decomposition]],
odd cohomology of a compact [[Kähler manifold]]
are always even-dimensional.
 
== Hypercomplex structure ==
Even-dimensional Hopf manifolds admit
[[Hypercomplex manifold|hypercomplex structure]].
The Hopf surface is the only compact [[hypercomplex manifold]] of quaternionic dimension 1 which is not [[hyperkähler manifold|hyperkähler]].
 
== References ==
*{{Citation | last1=Hopf | first1=Heinz | author1-link=Heinz Hopf | title=Studies and Essays Presented to R. Courant on his 60th Birthday, January 8, 1948 | publisher=Interscience Publishers, Inc., New York | id={{MathSciNet | id = 0023054}} | year=1948 | chapter=Zur Topologie der komplexen Mannigfaltigkeiten | pages=167–185}}
*{{eom|id=H/h110270|first=L. |last=Ornea}}
 
[[Category:Complex manifolds]]

Latest revision as of 04:07, 3 November 2014

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my website; at home std test (visit web site)