Nontrivial RR two-form field strength and SU(3)-structure

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We discuss how in the presence of a nontrivial RR two-form field strength and nontrivial dilaton the conditions of preserving supersymmetry on six-dimensional manifolds lead to generalized monopole and Killing spinor equations. We show that the manifold is Kähler in the ten-dimensional string frame if F₀⁽¹,¹⁾=0. We then determine explicitly the intrinsic torsion of the SU(3)-structure on six-manifolds that result via Kaluza-Klein reduction from seven-manifolds with G₂-structure of generic intrinsic torsion. Lastly we give explicitly the intrinsic torsion of the SU(3)-structure for an N=1 supersymmetric background in the presence of nontrivial RR two-form field strength and nontrivial dilaton.