Riemannian structures on higher order frame bundles over Riemannian manifolds

We describe all $\mathcal Mf_m$-natural operators $A\colon\mathcal Riem\to\mathcal Riem P^r$ transforming Riemannian structures $g$ on $m$-dimensional manifolds $M$ into Riemannian structures $A(g)$ on the $r$-th order frame bundle $P^rM=invJ_0^r(\mathbf R^m, M)$ over $M$.