In mathematics, a nearly Kähler manifold is an almost Hermitian manifold , with almost complex structure , such that the (2,1)-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. [1] 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 [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 , 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. [6] However, Foscolo and Haskins recently showed that and 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 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.
{{
cite journal}}
: CS1 maint: multiple names: authors list (
link)
In mathematics, a nearly Kähler manifold is an almost Hermitian manifold , with almost complex structure , such that the (2,1)-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. [1] 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 [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 , 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. [6] However, Foscolo and Haskins recently showed that and 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 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.
{{
cite journal}}
: CS1 maint: multiple names: authors list (
link)