Nearly Kähler manifold

In mathematics, a nearly Kähler manifold is an almost Hermitian manifold $M$, 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. 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 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]

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.^[5] 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.^[6] Nearly Kähler manifolds are an interesting class of manifolds admitting a metric connection with parallel totally antisymmetric torsion^[7]

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.

References

1. Franki Dillen and Leopold Verstraelen (eds.). Handbook of Differential Geometry, volume II. ISBN 978-0-444-82240-6. North Holland.
2. Chen, Bang-Yen (2011). Pseudo-Riemannian geometry, [delta]-invariants and applications. World Scientific. ISBN 978-981-4329-63-7.
3. "Nearly Kähler manifolds". J.Diff.Geometry 4. 1970. pp. 283–309.
4. Friedrich, Thomas; Grunewald, Ralf (1985). "On the first eigenvalue of the Dirac operator on 6-dimensional manifolds". Ann. Global Anal. Geom. 3. pp. 265–273.
5. Butruille, Jean-Baptiste (2005). "Classification of homogeneous nearly Kähler manifolds". Ann. Global Anal. Geom. 27. pp. 201–225.
6. Nagy, Paul-Andi (2002). "Nearly Kähler geometry and Riemannian foliations". Asian J. Math. 6. pp. 481–504.
7. Agricola, Ilka (2006). "The Srni lectures on non-integrable geometries with torsion". Archivum Mathematicum 42 (5). pp. 5–84. arXiv:math/0606705.