Curvature identities for principle T 1-bundles over almost Hermitian manifolds

We show that the identities R 1, R 2 and R 3 for an almost Hermitian structure S on the base of the canonical principal T 1-bundle are equivalent to their contact analogs for the induced almost contact metric structure S # on the total space of this bundle. We prove that the canonical connection of the canonical principal T 1-bundle over a Hermitian or a quasi-Kähler manifold of class R 3 is normal. We also prove that that the canonical connection of the canonical principal T 1-bundle over a Vaisman-Gray manifold M of class R 3 is normal if and only if the Lee vector of M belongs to the center of the adjoint K-algebra.