  We study almost Kaehler manifolds whose curvature tensor satisfies the third
curvature condition of Gray. We show that the study of manifolds within this
class reduces to the study of a subclass having the property that the torsion
of the first canonical Hermitian connection has the simplest possible algebraic
form. This allows to understand the structure of the Kaehler nullity of an
almost Kaehler manifold with parallel torsion.
