Nearly Kähler manifold explained

M , with almost complex structure J ,such that the (2,1)-tensor \nabla J is skew-symmetric. So,(\nabla _{X} J)X =0

for every vector field

X on M .

In particular, a Kähler manifold is nearly Kähler. The converse is not true. For example, the nearly Kähler six-sphere

S ^{6} 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 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^[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 Killingspinors: Thomas Friedrich and Ralf Grunewald showed that a 6-dimensional Riemannian manifold admitsa 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 thatthese 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

S ^{6,CP} ^{3,} P (T CP _{2)} , and S ^{3 x} S ^{3} . 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 S ^{6} and S ^{3 x} S ^{3} 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 withparallel totally antisymmetric torsion.^[9]

Nearly Kähler manifolds should not be confused with almost Kähler manifolds.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 essentially exclusive: an almost Hermitian manifold is both nearly Kähler and almost Kahler if and only if it is Kähler.

Notes and References

1. Book: Handbook of Differential Geometry. II . 978-0-444-82240-6. Franki Dillen . Leopold Verstraelen. North Holland.
2. Book: Chen, Bang-Yen. Pseudo-Riemannian geometry, [delta]-invariants and applications. World Scientific. 2011. 978-981-4329-63-7.
3. Nearly Kähler manifolds. J. Differ. Geom. . 4 . 1970 . 283–309. 10.4310/jdg/1214429504. Gray. Alfred. 3. free.
4. Friedrich . Thomas . Grunewald . Ralf. On the first eigenvalue of the Dirac operator on 6-dimensional manifolds. Ann. Global Anal. Geom. . 3 . 1985 . 3 . 265–273. 10.1007/BF00130480 . 120431819 .
5. Bär, Christian (1993) Real Killing spinors and holonomy. Comm. Math. Phys. 154, 509–521.
6. Butruille, Jean-Baptiste. Classification of homogeneous nearly Kähler manifolds. Ann. Global Anal. Geom. . 27 . 2005 . 201–225. 10.1007/s10455-005-1581-x. 118501746.
7. Foscolo, Lorenzo and Haskins, Mark . New G₂-holonomy cones and exotic nearly Kähler structures on S⁶ and S³ x S³ . Ann. of Math. . Series 2 . 185. 2017. 1 . 59–130. 10.4007/annals.2017.185.1.2 . 1501.07838 .
8. Nagy, Paul-Andi. Nearly Kähler geometry and Riemannian foliations. Asian J. Math. . 6 . 2002 . 3. 481–504. 10.4310/AJM.2002.v6.n3.a5. 117065633. free.
9. Agricola. Ilka. Ilka Agricola . The Srni lectures on non-integrable geometries with torsion. Archivum Mathematicum . 2006. 42 . 5. 5–84. math/0606705. 2006math......6705A.