Almost Hermitian 6-Manifolds Revisited

Details Almost Hermitian 6-Manifolds Revisited Almost Hermitian 6-Manifolds Revisited

2004-03-08

arxiv.org/abs/math/0403131v2

J.Geom.Phys. 53 (2005) 1-30

Mathematics

Differential Geometry

26 pages, revised version

Scientific paper

10.1016/j.geomphys.2004.04.009

A Theorem of Kirichenko states that the torsion 3-form of the characteristic connection of a nearly K\"ahler manifold is parallel. On the other side, any almost hermitian manifold of type $\mathrm{G}_1$ admits a unique connection with totally skew symmetric torsion. In dimension six, we generalize Kirichenko's Theorem and we describe almost hermitian $\mathrm{G}_1$-manifolds with parallel torsion form. In particular, among them there are only two types of $\mathcal{W}_3$-manifolds with a non-abelian holonomy group, namely twistor spaces of 4-dimensional self-dual Einstein manifolds and the invariant hermitian structure on the Lie group $\mathrm{SL}(2, \C)$. Moreover, we classify all naturally reductive hermitian $\mathcal{W}_3$-manifolds with small isotropy group of the characteristic torsion.

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