Let $G$ be a bundle functor of order $(r,s,q)$, $s\ge r\le q$, on the category $ℱ{ℳ}_{m,n}$ of $(m,n)$-dimensional fibered manifolds and local fibered diffeomorphisms. Given a general connection $\Gamma$ on an $ℱ{ℳ}_{m,n}$-object $Y\to M$ we construct a general connection $𝒢(\Gamma ,\lambda ,\Lambda )$ on $GY\to Y$ be means of an auxiliary $q$-th order linear connection $\lambda$ on $M$ and an $s$-th order linear connection $\Lambda$ on $Y$. Then we construct a general connection $𝒢(\Gamma ,{\nabla }_1,{\nabla }_2)$ on $GY\to Y$ by means of auxiliary classical linear connections ${\nabla }_1$ on $M$ and ${\nabla }_2$ on $Y$. In the case $G=J^1$ we determine all general connections $𝒟(\Gamma ,\nabla )$ on $J^{1}Y\to Y$ from...

This article deals with vector valued differential forms on $C^{\infty }$-manifolds. As a generalization of the exterior product, we introduce an operator that combines $Hom({⨂}^s\left(W\right),Z)$-valued forms with $Hom({⨂}^s\left(V\right),W)$-valued forms. We discuss the main properties of this operator such as (multi)linearity, associativity and its behavior under pullbacks, push-outs, exterior differentiation of forms, etc. Finally we present applications for Lie groups and fiber bundles.