Nearly Kähler manifold

In mathematics, a nearly Kähler manifold is an almost Hermitian manifold [math]\displaystyle{ M }[/math], with almost complex structure [math]\displaystyle{ J }[/math], such that the (2,1)-tensor [math]\displaystyle{ \nabla J }[/math] is skew-symmetric. So,

[math]\displaystyle{ (\nabla_X J)X =0 }[/math]

for every vector field [math]\displaystyle{ X }[/math] on [math]\displaystyle{ M }[/math].

In particular, a Kähler manifold is nearly Kähler. The converse is not true. For example, the nearly Kähler six-sphere [math]\displaystyle{ S^6 }[/math] is an example of a nearly Kähler manifold that is not Kähler. The familiar almost complex structure on the six-sphere is not induced by a complex atlas on [math]\displaystyle{ 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 and then by Alfred Gray from 1970 on. 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 and Ralf Grunewald showed that a 6-dimensional Riemannian manifold admits a Riemannian Killing spinor if and only if it is nearly Kähler. This was later given a more fundamental explanation by Christian Bär, who pointed out that these are exactly the 6-manifolds for which the corresponding 7-dimensional Riemannian cone has holonomy G₂.

The only compact simply connected 6-manifolds known to admit strict nearly Kähler metrics are [math]\displaystyle{ S^6,\mathbb{C}\mathbb{P}^3, \mathbb{P}(T\mathbb{CP}_2) }[/math], and [math]\displaystyle{ S^3\times S^3 }[/math]. Each of these admits such a unique nearly Kähler metric that is also homogeneous, and these examples are in fact the only compact homogeneous strictly nearly Kähler 6-manifolds. However, Foscolo and Haskins recently showed that [math]\displaystyle{ S^6 }[/math] and [math]\displaystyle{ S^3\times S^3 }[/math] also admit strict nearly Kähler metrics that are not homogeneous.

Bär's observation about the holonomy of Riemannian cones might seem to indicate that the nearly-Kähler condition is most natural and interesting in dimension 6. This actually borne out by a theorem of Nagy, who proved that any strict, complete nearly Kähler manifold is locally a Riemannian product of homogeneous nearly Kähler spaces, twistor spaces over quaternion-Kähler manifolds, and 6-dimensional nearly Kähler manifolds.

Nearly Kähler manifolds are also an interesting class of manifolds admitting a metric connection with parallel totally antisymmetric torsion.

Nearly Kähler manifolds should not be confused with almost Kähler manifolds. An almost Kähler manifold [math]\displaystyle{ M }[/math] is an almost Hermitian manifold with a closed Kähler form: [math]\displaystyle{ d\omega = 0 }[/math]. The Kähler form or fundamental 2-form [math]\displaystyle{ \omega }[/math] is defined by

[math]\displaystyle{ \omega(X,Y) = g(JX,Y), }[/math]

where [math]\displaystyle{ g }[/math] is the metric on [math]\displaystyle{ M }[/math]. The nearly Kähler condition and the almost Kähler condition are essentially exclusive: an almost Hermitian manifold is both nearly Kähler and almost Kahler if and only if it is Kähler.