Torsion and curvature of ACYT and AHKT 8-manifold

Abstract

We observe that on a compact almost Hermitian 8 manifold with totally skew-symmetric Nijenhuis tensor parallel with respect to the unique almost hermitian connection with torsion three form if the three Ricci tensors of this connection vanish then the torsion 3-form is closed. Consequently, on a compact complex 8-manifold if the three Ricci tensors of the Strominger-Bismut connection vanish then its torsion is closed, i.e. the space is pluriclosed. We also deduce that a compact ACYT 8-manifold with paralle Nijenhuis tensor with respect to the torsion connection has closed torsion if and only if the trace of the exterior derivative of the torsion vanishes. It is shown that a compact ACYT 8-manifold wich Nijenhuis tensor is parallel with respect to the torsion connection is a Ricci flat $SU(4)$ instanton exactly when the torsion 3-form is parallel with respect to the torsion and to the Levi-Civita connections simultaneously. We consider an almost hyperhermitian 8-manifold with an $Sp(2)$ structure admitting linear connection preserving the $Sp(2)$ structure and having totally skew symmetric torsion and call it AHKT manifold. The exact conditions an almost hyperhermitian 8-manifold to be an AHKT are presented and it is shown that an AHKT 8-manifold is HKT exactly when the three Lee forms coincide