In mathematics, a nearly Kähler manifold is an almost Hermitian manifold
M
, with almost complex structure
J
, such that the (2,1)-tensor
∇
J
is skew-symmetric. So,
(
∇
X
J
)
X
=
0
for every vector field
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
/
S
U
(
3
)
,
S
p
(
2
)
/
(
S
U
(
2
)
×
U
(
1
)
)
,
S
U
(
3
)
/
(
U
(
1
)
×
U
(
1
)
)
,
S
3
×
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
ω
=
0
. The Kähler form or fundamental 2-form
ω
is defined by
ω
(
X
,
Y
)
=
g
(
J
X
,
Y
)
,
where
g
is the metric on
M
. The nearly Kähler condition and the almost Kähler condition are mutually exclusive.
