In mathematics, a nearly Kähler manifold is an almost Hermitian manifold, with almost complex structure, such that the -tensor is skew-symmetric. So, for every vector field on. In particular, a Kähler manifold is nearly Kähler. The converse is not true. For example, the nearly Kähler six-sphere 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. 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 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 spinorif 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, and. 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 and 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 is an almost Hermitian manifold with a closed Kähler form: . The Kähler form or fundamental 2-form is defined by where is the metric on. 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.
