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

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

for every vectorfield $X$ on $M$.

In particular, a Kähler manifold is nearly Kähler. The converse is not true. The nearly Kähler six-sphere $S^6$ 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 $S^6$. 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 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. The only known 6-dimensional strict nearly Kähler manifolds are: $S^6=G_2/SU(3),Sp(2)/SU(2)\times U(1),SU(3)/U(1)\times U(1),S^3\times S^3$. In fact, these are the only homogeneous nearly Kähler manifolds in dimension six. 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. Nearly Kähler manifolds are an interesting class of manifolds admitting a metric connection with parallel totally antisymmetric torsion

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

$\omega (X,Y)=g(JX,Y),$

where $g$ is the metric on $M$. The nearly Kähler condition and the almost Kähler condition are mutually exclusive.

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