Strong Kaehler with torsion structures from almost contact manifolds

For an almost contact metric manifold $N$, we find conditions for which either the total space of an $S^1$-bundle over $N$ or the Riemannian cone over $N$ admits a strong K\"ahler with torsion (SKT) structure. In this way we construct new 6-dimensional SKT manifolds. Moreover, we study the geometric structure induced on a hypersurface of an SKT manifold, and use such structures to construct new SKT manifolds via appropriate evolution equations. Hyper-K\"ahler with torsion (HKT) structures on the total space of an $S^1$-bundle over manifolds with three almost contact structures are also studied.