  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.

Differential Geometry