Nearly Kähler manifold

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

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

for every vector field ${\displaystyle X}$ on ${\displaystyle M}$.

In particular, a Kähler manifold is nearly Kähler. The converse is not true. For example, the nearly Kähler six-sphere ${\displaystyle S^{6}}$ 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 ${\displaystyle S^{6}}$. 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 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 ${\displaystyle S^{6},\mathbb {C} \mathbb {P} ^{3},\mathbb {P} (T\mathbb {CP} _{2})}$, and ${\displaystyle S^{3}\times S^{3}}$. 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 ${\displaystyle S^{6}}$ and ${\displaystyle S^{3}\times S^{3}}$ 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 with parallel totally antisymmetric torsion.[9]

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

${\displaystyle \omega (X,Y)=g(JX,Y),}$

where ${\displaystyle g}$ is the metric on ${\displaystyle M}$. 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.

References

1. Franki Dillen; Leopold Verstraelen (eds.). Handbook of Differential Geometry. II. North Holland. ISBN 978-0-444-82240-6.
2. Chen, Bang-Yen (2011). Pseudo-Riemannian geometry, [delta]-invariants and applications. World Scientific. ISBN 978-981-4329-63-7.
3. Gray, Alfred (1970). "Nearly Kähler manifolds". J.Diff.Geometry. 4 (3): 283–309. doi:10.4310/jdg/1214429504.
4. Friedrich, Thomas; Grunewald, Ralf (1985). "On the first eigenvalue of the Dirac operator on 6-dimensional manifolds". Ann. Global Anal. Geom. 3 (3): 265–273. doi:10.1007/BF00130480. S2CID 120431819.
5. Bär, Christian (1993) Real Killing spinors and holonomy. Comm. Math. Phys. 154, 509–521.
6. Butruille, Jean-Baptiste (2005). "Classification of homogeneous nearly Kähler manifolds". Ann. Global Anal. Geom. 27: 201–225. doi:10.1007/s10455-005-1581-x. S2CID 118501746.
7. Foscolo, Lorenzo and Haskins, Mark (2017). "New G₂-holonomy cones and exotic nearly Kähler structures on S⁶ and S³ x S³". Ann. of Math. 2. 185 (1): 59–130. arXiv:1501.07838. doi:10.4007/annals.2017.185.1.2.CS1 maint: multiple names: authors list (link)
8. Nagy, Paul-Andi (2002). "Nearly Kähler geometry and Riemannian foliations". Asian J. Math. 6 (3): 481–504. doi:10.4310/AJM.2002.v6.n3.a5. S2CID 117065633.
9. Agricola, Ilka (2006). "The Srni lectures on non-integrable geometries with torsion". Archivum Mathematicum. 42 (5): 5–84. arXiv:math/0606705. Bibcode:2006math......6705A.