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We show that on an hyper-Kähler with torsion (HKT) manifold the holonomy of the Obata connection is contained inif and only if the Lee form is an exact 1-form. As an application, we show compact HKT manifolds with holomorphically trivial canonical bundle that are not balanced. A simple criterion for nonexistence of HKT metric on hypercomplex manifold is given in terms of the Ricci-type tensors of the Obata connection.

HKT Manifolds with Holonomy SL( n , H )