Abstract
An Hermitian metric is said to be Calabi-Yau with torsion (CYT in short) if it has vanishing
first Bismut–Ricci curvature, namely a natural Ricci-type curvature which coincides
with the usual Ricci form when the metric is Kähler. When a Hermitian structure is SKT
and CYT, then it is a static point of the puriclosed flow, motivating a huge interest in finding
explicit non-trivial examples, where by non trivial we mean not diffeomorphic to a global
product of a Kähler-Ricci flat manifold and a Bismut flat space. Moreover, up to now, no
non-trivial examples are known. In a work in progress with A. Fino and G. Grantcharov
we investigate SKT and CYT structures with parallel Bismut torsion. We first characterize
the universal cover of such manifolds in the compact case and we use this characterization
to construct the first known non-trivial example of CYT and SKT manifold. It can be described
as a mapping torus of a product of a compact Kähler Ricci flat manifold with the
3-sphere or, alternatively, as a total space of an holomorphic fibration with Kähler fibre over
the Hopf Surface. The existence of generalized Kähler structures is also investigated.
