Higher order Utiyama-like theorem

In this paper we prove higher order version of the Utiyama-like theorem. To prove the Utiyama-like theorem in order r ≥ 2 we have to use auxiliary classical connections on base manifolds. We prove that any natural (invariant) operator of order r for principal connections on principal G-bundles and for classical connections on base manifolds with values in a (1, 0)-order G-gauge-natural bundle factorizes through curvature tensors of both connections and their co-variant differentials, where the covariant differential of curvature tensors of principal connections is considered with respect to both connections.