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 ${\mathrm{G}}_1$ 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 $\mathrm{S}\mathrm{L}(2,\text{C})$. Moreover, we classify all naturally reductive hermitian ${\mathcal{W}}_3$-manifolds with small isotropy group of the characteristic torsion.