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.

We prove that any bundle functor F:ℱol → ℱℳ on the category ℱ olof all foliated manifolds without singularities and all leaf respecting maps is of locally finite order.

Let $M$ be a smooth manifold, $A$ a local algebra in sense of André Weil, $M^A$ the manifold of near points on $M$ of kind $A$ and $𝔛\left(M^A\right)$ the module of vector fields on $M^A$. We give a new definition of vector fields on $M^A$ and we show that $𝔛\left(M^A\right)$ is a Lie algebra over $A$. We study the cohomology of $A$-differential forms. Résumé. On considère $M$ une variété différentielle, $A$ une algèbre locale au sens d’André Weil, $M^A$ la variété des points proches de $M$ d’espèce $A$ et $𝔛\left(M^A\right)$ le module des champs de vecteurs sur $M^A$. On donne une nouvelle...

Among all $C^{\infty }$-algebras we characterize those which are algebras of $C^{\infty }$-functions on second countable Hausdorff $C^{\infty }$-manifolds.

We classify all $ℱ{ℳ}_{m,n}$-natural operators $:J²⟿J²V^A$ transforming second order connections Γ: Y → J²Y on a fibred manifold Y → M into second order connections $\left(\Gamma \right):V^{A}Y\to J²V^{A}Y$ on the vertical Weil bundle $V^{A}Y\to M$ corresponding to a Weil algebra A.