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.

In this paper we extend the variational calculus to fibered-fibered manifolds. Fibered-fibered manifolds are surjective fibered submersions π:Y → X between fibered manifolds. For natural numbers s ≥ r ≤ q with r ≥ 1 we define (r,s,q)th order Lagrangians on fibered-fibered manifolds π:Y → X as base-preserving morphisms $\lambda :J^{r,s,q}Y\to {\bigwedge }^{dimX}T*X$. Then similarly to the fibered manifold case we define critical fibered sections of Y. Setting p=max(q,s) we prove that there exists a canonical “Euler” morphism $\left(\lambda \right):J^{r+s,2s,r+p}Y\to *Y\otimes {\bigwedge }^{dimX}T*X$ of λ satisfying...

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.

We recall several different definitions of semiholonomic jet prolongations of a fibered manifold and use them to derive some interesting properties of prolongation of a first order connection to a third order semiholonomic connection.

The descriptions of Weil bundles, lifts of functions and vector fields are given. Actions of the automorphisms group of the Whitney sum of algebras of dual numbers on a Weil bundle of the first order are defined.

Let $\mu :FX\to X$ be a principal bundle of frames with the structure group ${\mathrm{Gl}}_n\left(ℝ\right)$. It is shown that the variational problem, defined by ${\mathrm{Gl}}_n\left(ℝ\right)$-invariant Lagrangian on $J^{r}FX$, can be equivalently studied on the associated space of connections with some compatibility condition, which gives us order reduction of the corresponding Euler-Lagrange equations.

The aim of the present work is to present a geometric formulation of higher order variational problems on arbitrary fibred manifolds. The problems of Engineering and Mathematical Physics whose natural formulation requires the use of second order differential invariants are classic, but it has been the recent advances in the theory of integrable non-linear partial differential equations and the consideration in Geometry of invariants of increasingly higher orders that has highlighted the interest...

We discuss frame bundles and canonical forms for geometries modeled on homogeneous spaces. Our aim is to introduce a geometric picture based on the non-holonomic jet bundles and principal prolongations as introduced in [Kolář, 71]. The paper has a partly expository character and we focus on very general aspects only. In the final section, various links to known results on the parabolic geometries are given briefly and some directions for further investigations are roughly indicated.

We investigate the category of product preserving bundle functors defined on the category of fibered fibered manifolds. We show a bijective correspondence between this category and a certain category of commutative diagrams on product preserving bundle functors defined on the category ℳ f of smooth manifolds. By an application of the theory of Weil functors, the latter category is considered as a category of commutative diagrams on Weil algebras. We also mention the relation with natural transformations...

We present a complete description of all product preserving bundle functors on the category ℱol of all foliated manifolds and their leaf respecting maps in terms of homomorphisms of Weil algebras.

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....

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.