Affine structure on Weil bundles

Abstract

For every $r$-th order Weil functor $T^{A}$, we introduce the underlying $k$-th order Weil functors $T^{A_{k}}$, $k = 1, \ldots, r-1$. We deduce that $T^{A}M \to T^{A_{r-1}} M$ is an affine bundle for every manifold $M$. Generalizing the classical concept of contact element by C. Ehresmann, we define the bundle $\mathcal{k}T^{A}M$ of contact elements of type $A$ on $M$ and we describe some affine properties of this bundle.