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Abstract

We describe all natural operators $\mathcal{A}$ lifting nowhere vanishing vector fields $X$ on $m$-dimensional manifolds $M$ to vector fields $\mathcal{A}(X)$ on the $r$th order frame bundle $L^rM=\mathop{\rm inv} J^r_0(\mathbb{R}^m, M)$ over~$M$. Next, we describe all natural operators $\mathcal{A}$ lifting vector fields $X$ on $m$-manifolds $M$ to vector fields on $L^rM$. In both cases we deduce that the spaces of all operators $\mathcal{A}$ in question form free $(m(C^{m+r}_r-1)+1)$-dimensional modules over algebras of all smooth maps $J^{r-1}_0\widetilde T\mathbb{R}^m\to\mathbb{R}$ and $J^{r-1}_0T\mathbb{R}^m\to\mathbb{R}$ respectively, where $C^n_k={n!/(n-k)!k!}$. We explicitly construct bases of these modules. In particular, we find that the vector space over $\mathbb{R}$ of all natural linear operators lifting vector fields $X$ on $m$-manifolds $M$ to vector fields on $L^rM$ is $(m^2C^{m+r-1}_{r-1}(C^{m+r}_r-1)+1)$-dimensional.