Almost Hermitian 6-manifolds revisited

Abstract

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.