Description of jet like functors on vector bundles by means of module bundle functors on the bases

Abstract

Let  $\mathcal C$ be an admissible category over manifolds and $\mathcal{VB_C}$ be the category of vector bundles with bases being $\mathcal{C}$-objects and vector bundle maps with base maps being $\mathcal{C}$-maps. Assume that any $\mathcal{C}$-map is a local isomorphism.  We describe all jet like functors (i.e. fiber product preserving gauge bundle functors) of order $r$ on $\mathcal{VB_C}$ be means of $J^r(-,\mathbf{R})$-module bundle functors on $\mathcal{C}$. Then we describe all jet like functors of vertical type of order $r$ on $\mathcal{VB_C}$ be means of vector bundle functors on $\mathcal{C}$ of order $r$. As an application we classify all jet like functors on $\mathcal{VB}_m$. Finally we determine all $J^r(M,\mathbfR})-module bundle structures  on the vector bundles  $J^rE\to M$ and $J^r_vE\to M$, where $E\to M$ is a vector bundle with $m$-dimensional basis and $m\geq 2$.