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.
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 . For p= max(q,s) there exists a canonical Euler morphism 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 . In the present paper, similarly...
We deduce further properties of connections on the functional bundle of all smooth maps between the fibers over the same base point of two fibered manifolds over the same base, which we introduced in [2]. In particular, we define the vertical prolongation of such a connection, discuss the iterated absolute differentiation by means of an auxiliary linear connection on the base manifold and prove the general Ricci identity.
Using Weil algebra techniques, we determine all finite dimensional homomorphic images of germs of foliation respecting maps.
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 a linear r-th order connection on the tangent bundle we characterize geometrically its integrability in the sense of the theory of higher order G-structures. Our main tool is a bijection between these connections and the principal connections on the r-th order frame bundle and the comparison of the torsions under both approaches.