Nearly Kähler manifold

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

${\displaystyle (\nabla _{X}J)X=0}$

for every vector field ${\displaystyle X}$ on ${\displaystyle M}$.

In particular, a Kähler manifold is nearly Kähler. The converse is not true. For example, the nearly Kähler six-sphere ${\displaystyle S^{6}}$ is an example of a nearly Kähler manifold that is not Kähler.^[1] The familiar almost complex structure on the six-sphere is not induced by a complex atlas on ${\displaystyle S^{6}}$. 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^[2] and then by Alfred Gray from 1970 on.^[3] 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.^[4] This was later given a more fundamental explanation ^[5] 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 ${\displaystyle S^{6},\mathbb {C} \mathbb {P} ^{3},\mathbb {P} (T\mathbb {CP} _{2})}$, and ${\displaystyle S^{3}\times S^{3}}$. 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.^[6] However, Foscolo and Haskins recently showed that ${\displaystyle S^{6}}$ and ${\displaystyle S^{3}\times S^{3}}$ also admit strict nearly Kähler metrics that are not homogeneous.^[7]

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.^[8]

Nearly Kähler manifolds are also an interesting class of manifolds admitting a metric connection with parallel totally antisymmetric torsion.^[9]

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

${\displaystyle \omega (X,Y)=g(JX,Y),}$

where ${\displaystyle g}$ is the metric on ${\displaystyle M}$. 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.