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Almost Hermitian 6-Manifolds Revisited
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 admits a unique connection
with totally skew symmetric torsion. In dimension six, we generalize
Kirichenko's Theorem and we describe almost hermitian -manifolds
with parallel torsion form. In particular, among them there are only two types
of -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 -manifolds with small
isotropy group of the characteristic torsion.Comment: 26 pages, revised versio
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