Functorial prolongations of some functional bundles

TL;DR

This paper explores two types of functorial prolongations of functional bundles between fibered manifolds, demonstrating the preservation of the Lie bracket of vector fields through these prolongations, supported by new results on Weil bundles.

Contribution

It introduces and analyzes two functorial prolongation methods for functional bundles, establishing the preservation of vector field brackets and providing new insights into Weil bundles.

Findings

Proves bracket preservation in both prolongation cases

Develops new results on finite dimensional Weil bundles

Enhances understanding of functorial properties in fibered manifolds

Abstract

We discuss two kinds of functorial prolongations of the functional bundle of all smooth maps between the fibers over the same base point of two fibered manifolds over the same base. We study the prolongation of vector fields in both cases and we prove that the bracket is preserved. Our proof is based on several new results concerning the finite dimensional Weil bundles.