Higher order connections and fields of geometric objects on manifolds depending on parameters

In the present paper we study manifolds depending on $N$ parameters, i.e. fibered manifolds $p\colon E\to U$, where $U\subset\mathbf R^N$ is an open subset in $\mathbf R^N$. To the Weil bundle $\widehat T^{\mathbf A}(E)$ we associate a sequence of principal $\mathbf A$-affine frame bundles of higher order, this makes it possible to consider fields of (vertical) differential geometric objects on $E$ as sections of the corresponding associated bundles. In particular, we construct the bundle of $\mathbf A$-affine connections on $E$. We construct also complete lifts of geometric objects from E to the Weil bundle $\widehat T^{\mathbf B}(E)$, where $\mathbf B$ is the Weil algebra of width $N$.