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 G1 admits a unique connection with totally skew-symmetric torsion. In dimension 6, we generalize Kirichenko's theorem and we describe almost Hermitian G1-manifolds with parallel torsion form. In particular, among them there are only two types of W3-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 W3-manifolds with small isotropy group of the characteristic torsion.