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

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