For example, the nearly Kähler six-sphere
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
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
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.
is an almost Hermitian manifold with a closed Kähler form: