Elements of general theory of infinitely prolonged underdetermined systems of ordinary differential equations are outlined and applied to the equivalence of one-dimensional constrained variational integrals. The relevant infinite-dimensional variant of Cartan’s moving frame method expressed in quite elementary terms proves to be surprisingly efficient in solution of particular equivalence problems, however, most of the principal questions of the general theory remains unanswered. New concepts of...

We classify all bundle functors $G$ admitting natural operators transforming connections on a fibered manifold $Y\to M$ into connections on $GY\to M$. Then we solve a similar problem for natural operators transforming connections on $Y\to M$ into connections on $GY\to Y$.

We present a complete description of all fiber product preserving gauge bundle functors F on the category ${}_m$ of vector bundles with m-dimensional bases and vector bundle maps with local diffeomorphisms as base maps. Some corollaries of this result are presented.

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.

A Goursat structure on a manifold of dimension $n$ is a rank two distribution $𝒟$ such that dim ${𝒟}^{\left(i\right)}=i+2$, for $0\le i\le n-2$, where ${𝒟}^{\left(i\right)}$ denote the elements of the derived flag of $𝒟$, defined by ${𝒟}^{\left(0\right)}=𝒟$ and ${𝒟}^{(i+1)}={𝒟}^{\left(i\right)}+[{𝒟}^{\left(i\right)},{𝒟}^{\left(i\right)}]$. Goursat structures appeared first in the work of von Weber and Cartan, who have shown that on an open and dense subset they can be converted into the so-called Goursat normal form. Later, Goursat structures have been studied by Kumpera and Ruiz. In the paper, we introduce a new local invariant for Goursat structures, called...

A Goursat structure on a manifold of dimension n is a rank two distribution Ɗ such that dim Ɗ(i) = i + 2, for 0 ≤ i ≤ n-2, where Ɗ(i) denote the elements of the derived flag of Ɗ, defined by Ɗ(0) = Ɗ and Ɗ(i+1) = Ɗ(i) + [Ɗ(i),Ɗ(i)] . Goursat structures appeared first in the work of von Weber and Cartan, who have shown that on an open and dense subset they can be converted into the so-called Goursat normal form. Later, Goursat structures have been studied by Kumpera and Ruiz. In the paper, we introduce...

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.

The group of real analytic diffeomorphisms of a real analytic manifold is a rich group. It is dense in the group of smooth diffeomorphisms. Herman showed that for the $n$-dimensional torus, its identity component is a simple group. For $U\left(1\right)$ fibered manifolds, for manifolds admitting special semi-free $U\left(1\right)$ actions and for 2- or 3-dimensional manifolds with nontrivial $U\left(1\right)$ actions, we show that the identity component of the group of real analytic diffeomorphisms is a perfect group.

A fibered-fibered manifold is a surjective fibered submersion π: Y → X between fibered manifolds. For natural numbers s ≥ r ≤ q an (r,s,q)th order Lagrangian on a fibered-fibered manifold π: Y → X is a base-preserving morphism $\lambda :J^{r,s,q}Y\to {\bigwedge }^{dimX}T*X$. For p= max(q,s) there exists a canonical Euler morphism $\left(\lambda \right):J^{r+s,2s,r+p}Y\to *Y\otimes {\bigwedge }^{dimX}T*X$ satisfying a decomposition property similar to the one in the fibered manifold case, and the critical fibered sections σ of Y are exactly the solutions of the Euler-Lagrange equation $\left(\lambda \right)\circ j^{r+s,2s,r+p}\sigma =0$. In the present paper, similarly...