On the Constant Type of Almost Hermitian Manifolds

We suggest a most natural generalization of the notion of constant type for nearly Kählerian manifolds introduced by A. Gray to arbitrary almost Hermitian manifolds. We prove that the class of almost Hermitian manifolds of zero constant type coincides with the class of Hermitian manifolds. We show that the class of $G_1$-manifolds of zero constant type coincides with the class of 6-dimensional $G_1$-manifolds with a non-integrable structure. Finally, we prove that the class of normal $G_2$-manifolds of nonzero constant type coincides with the class of 4-dimensional $G_2$-manifolds with a nonintegrable structure.