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In mathematics, a '''nearly Kähler manifold''' is an [[almost Hermitian manifold]] <math>M</math>, with [[almost complex structure]] <math> J</math>,
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such that the (2,1)-tensor <math>\nabla J </math> is [[skew-symmetric]]. So,
 
:<math> (\nabla_X J)X =0 \, </math>
 
for every vectorfield <math>X</math> on <math>M</math>.
 
In particular, a [[Kähler manifold]] is nearly Kähler. The converse is not true.
The nearly Kähler six-sphere <math>S^6</math> is an example of a nearly Kähler manifold that is not Kähler.<ref>
{{cite book
|title=Handbook of Differential Geometry, volume II. ISBN 978-0-444-82240-6
|editors=Franki Dillen and Leopold Verstraelen
|publisher=North Holland}}</ref> The familiar almost complex structure on the six-sphere is not induced by a complex atlas on <math>S^6</math>.
Usually, non Kählerian nearly Kähler manifolds are called "strict nearly Kähler manifolds".  
Nearly Kähler manifolds, also known as almost Tachibana manifolds, were studied by Shun-ichi Tachibana in 1959<ref>
{{cite book|first=Bang-Yen|last=Chen|title=Pseudo-Riemanniann geometry, [delta]-invariants and applications|publisher=World Scientific|year=2011|isbn=978-981-4329-63-7}}</ref> and then by [[Alfred Gray (mathematician)|Alfred Gray]] from 1970 on.<ref>{{cite article
title=Nearly Kähler manifolds
journal=J.Diff.Geometry 4 (1970), 283-309.}}</ref>
For example, it was proved that any 6-dimensional strict nearly Kähler manifold is an [[Einstein manifold]] and has vanishing first Chern class
(in particular, this implies spin).  
In the 1980s, strict nearly Kähler manifolds obtained a lot of consideration because of their relation to [[Killing
spinors]]: [[Thomas Friedrich (Mathematiker)|Thomas Friedrich]] and Ralf Grunewald showed that a 6-dimensional Riemannian manifold admits
a Riemannian Killing spinor if and only if it is nearly Kähler.<ref>{{cite article
author=Friedrich, Thomas and Grunewald, Ralf
title=On the first eigenvalue of the Dirac operator on 6-dimensional manifolds
journal=Ann. Global Anal. Geom. 3 (1985), 265-273.}}</ref>
The only known 6-dimensional strict nearly Kähler manifolds are: <math>S^6=G_2/SU(3), Sp(2)/SU(2)\times U(1), SU(3)/U(1)\times U(1), S^3\times S^3</math>. In fact, these are the only homogeneous nearly Kähler manifolds in dimension six.<ref>{{cite article
author=Butruille, Jean-Baptiste
title= Classification of homogeneous nearly Kähler manifolds
journal=Ann. Global Anal. Geom.27 (2005), 201-225.}}</ref>
In applications, it is apparent that nearly Kähler manifolds are most interesting in dimension 6; in 2002. Paul-Andi Nagy
proved that indeed any strict and complete nearly Kähler manifold is locally a Riemannian product of homogeneous nearly Kähler spaces, twistor spaces over Kähler manifolds and 6-dimensional nearly Kähler manifolds.<ref>{{cite article
author=Nagy, Paul-Andi
title=Nearly Kähler geometry and Riemannian foliations
journal=Asian J. Math.6 (2002), 481-504.}}</ref>
Nearly Kähler manifolds are an interesting class of manifolds admitting a metric connection with
parallel totally antisymmetric torsion<ref>{{cite article
author=Agricola, Ilka
title=The Srni lectures on non-integrable geometries with torsion
journal=Arch. Math 42, 5–84.}}</ref>
 
A nearly Kähler manifold should not be confused with an [[almost Kähler manifold]].
An almost Kähler manifold <math>M</math> is an almost Hermitian manifold with a closed [[Kähler manifold|Kähler form]]:
<math>d\omega = 0</math>. The Kähler form or fundamental 2-form <math>\omega</math> is defined by
 
:<math>\omega(X,Y) = g(JX,Y), \, </math>
 
where <math>g</math> is the metric on <math>M</math>. The  nearly Kähler condition and the almost Kähler condition are mutually exclusive.
 
==References==
{{Reflist}}
 
{{DEFAULTSORT:Nearly Kahler manifold}}
[[Category:Topology]]
[[Category:Differential geometry]]
[[Category:Manifolds]]

Latest revision as of 15:42, 21 November 2014

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