We describe the fundamental properties of the infinitesimal actions related with functorial prolongations of principal and associated bundles with respect to fiber product preserving bundle functors. Our approach is essentially based on the Weil algebra technique and an original concept of weak principal bundle.

We describe some general geometric properties of the fiber product preserving bundle functors. Special attention is paid to the vertical Weil bundles. We discuss namely the flow natural maps and the functorial prolongation of connections.

Let $Y\to M$ be a fibred manifold with $m$-dimensional base and $n$-dimensional fibres and $E\to M$ be a vector bundle with the same base $M$ and with $n$-dimensional fibres (the same $n$). If $m\ge 2$ and $n\ge 3$, we classify all canonical constructions of a classical linear connection $A(\Gamma ,\Lambda ,\Phi ,\Delta )$ on $Y$ from a system $(\Gamma ,\Lambda ,\Phi ,\Delta )$ consisting of a general connection $\Gamma$ on $Y\to M$, a torsion free classical linear connection $\Lambda$ on $M$, a vertical parallelism $\Phi :Y{\times }_{M}E\to VY$ on $Y$ and a linear connection $\Delta$ on $E\to M$. An example of such $A(\Gamma ,\Lambda ,\Phi ,\Delta )$ is the connection $(\Gamma ,\Lambda ,\Phi ,\Delta )$ by I. Kolář.

First we deduce some general results on the covariant form of the natural transformations of Weil functors. Then we discuss several geometric properties of these transformations, special attention being paid to vector bundles and principal bundles.

For every bundle functor we introduce the concept of subordinated functor. Then we describe subordinated functors for fiber product preserving functors defined on the category of fibered manifolds with $m$-dimensional bases and fibered manifold morphisms with local diffeomorphisms as base maps. In this case we also introduce the concept of the underlying functor. We show that there is an affine structure on fiber product preserving functors.

We characterize Weilian prolongations of natural bundles from the viewpoint of certain recent general results. First we describe the iteration $F\left(EM\right)$ of two natural bundles $E$ and $F$. Then we discuss the Weilian prolongation of an arbitrary associated bundle. These two auxiliary results enables us to solve our original problem.

It is well known that the concept of holonomic $r$-jet can be geometrically characterized in terms of the contact of individual curves. However, this is not true for the semiholonomic $r$-jets, [5], [8]. In the present paper, we discuss systematically the semiholonomic case.

Let 𝓟𝓑 be the category of principal bundles and principal bundle homomorphisms. We describe completely the product preserving gauge bundle functors (ppgb-functors) on 𝓟𝓑 and their natural transformations in terms of the so-called admissible triples and their morphisms. Then we deduce that any ppgb-functor on 𝓟𝓑 admits a prolongation of principal connections to general ones. We also prove a "reduction" theorem for prolongations of principal connections into principal ones by means of Weil functors....

A complete description is given of all product preserving gauge bundle functors F on vector bundles in terms of pairs (A,V) consisting of a Weil algebra A and an A-module V with $di{m}_{ℝ}\left(V\right)<\infty$. Some applications of this result are presented.

Let $F$ be a natural bundle. We introduce the geometrical construction transforming two general connections into a general connection on the $F$-vertical bundle. Then we determine all natural operators of this type and we generalize the result by IK̇olář and the second author on the prolongation of connections to $F$-vertical bundles. We also present some examples and applications.

In this paper, $M$ denotes a smooth manifold of dimension $n$, $A$ a Weil algebra and $M^A$ the associated Weil bundle. When $(M,{\omega }_M)$ is a Poisson manifold with $2$-form ${\omega }_M$, we construct the $2$-Poisson form ${\omega }_{M^A}^A$, prolongation on $M^A$ of the $2$-Poisson form ${\omega }_M$. We give a necessary and sufficient condition for that $M^A$ be an $A$-Poisson manifold.

We prove the (first) reduction theorem for general and classical connections, i.e. we prove that any natural operator of a general connection Γ on a fibered manifold and a classical connection Λ on the base manifold can be expressed as a zero order operator of the curvature tensors of Γ and Λ and their appropriate derivatives.

We consider the problem of prolongating general connections on arbitrary fibered manifolds with respect to a product preserving bundle functor. Our main tools are the theory of Weil algebras and the Frölicher-Nijenhuis bracket.