Almost Hermitian 6-manifolds revisited

A theorem of Kirichenko states that the torsion 3-form of the characteristic connection of a nearly Kähler manifold is parallel. On the other side, any almost Hermitian manifold of type G 1 admits a unique connection with totally skew-symmetric torsion. In dimension 6, we generalize Kirichenko’s the...

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Bibliographic Details
Published in:Journal of geometry and physics 2005, Vol.53 (1), p.1-30
Main Authors: Alexandrov, Bogdan, Friedrich, Thomas, Schoemann, Nils
Format: Article
Language:English
Subjects:
Almost Hermitian manifolds
Connections with torsion
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Summary:A theorem of Kirichenko states that the torsion 3-form of the characteristic connection of a nearly Kähler manifold is parallel. On the other side, any almost Hermitian manifold of type G 1 admits a unique connection with totally skew-symmetric torsion. In dimension 6, we generalize Kirichenko’s theorem and we describe almost Hermitian G 1-manifolds with parallel torsion form. In particular, among them there are only two types of W 3 -manifolds with a non-Abelian holonomy group, namely twistor spaces of four-dimensional self-dual Einstein manifolds and the invariant Hermitian structure on the Lie group SL(2, C) . Moreover, we classify all naturally reductive Hermitian W 3 -manifolds with small isotropy group of the characteristic torsion.
ISSN:0393-0440
1879-1662
DOI:10.1016/j.geomphys.2004.04.009