HKT manifolds with holonomy SL(n,H)

Abstract: We show that on an HKT manifold the holonomy of the Obata connection is contained in SL(n,H) if and only if the Lee form is an exact one form. As an application, we show compact HKT manifolds with holomorphically trivial canonical bundle which are not balanced. A simple criterion for non-existence of HKT metric on hypercomplex manifold is given in terms of the Ricci-type tensors of the Obata connection.