Lifting to the $r$-frame bundle by means of connections
Let $m$ and $r$ be natural numbers and let $P^r:\mathcal M f_m\to\mathcal F\mathcal M$ be the $r$th order frame bundle functor. Let $F:\mathcal M f_m\to\mathcal F\mathcal M$ and $G:\mathcal M f_k\to\mathcal F\mathcal M$ be natural bundles, where $k=\dim (P^r\mathbb R^m)$. We describe all $\mathcal M f_m$-natural operators $A$ transforming sections $\sigma$ of $FM\to M$ and classical linear connections $\nabla$ on $M$ into sections $A(\sigma,\nabla)$ of $G(P^rM)\to P^rM$. We apply this general classification result to many important natural bundles $F$ and $G$ and obtain many particular classifications.