A type of nearly parallel G-structures

We study properties of a certain symmetric tensor r induced by the intrinsic torsion of a Riemannian G–structure. This tensor naturally arises in the context of nearly Kähler manifolds and is parallel with respect to the canonical Hermitian connection. In general, we call a G-structure a second order parallel if ${\nabla }^{G}r=0$ for a minimal G–connection ${\nabla }^G$. We show correlation with harmonicity of a G–structure and with G–structures with parallel torsion. An example of second order parallel G–structure is, apart from nearly Kähler manifolds and nearly parallel $G_2$ structures, an α-Kenmotsu manifold.