Affine Structures on Jet and Weil Bundles

TL;DR

This paper investigates conditions under which Weil algebra morphisms induce affine structures on Weil bundles and Jet spaces, revealing algebraic criteria for these geometric structures to exist.

Contribution

It identifies specific algebraic conditions, such as surjectivity and null square kernels, that guarantee affine bundle structures on Weil and Jet bundles.

Findings

Affine structures arise only when Weil algebra morphisms are surjective with null square kernels.

The affine structure on Weil bundles can be transferred to Jet spaces under certain conditions.

An algebraic characterization of the automorphism groups of Weil algebras is provided.

Abstract

Weil algebra morphism induce natural transformations between Weil bundles. In some well known cases, a natural transformation is endowed with a canonical structure of affine bundle. We show that this structure arises only when the Weil algebra morphism is surjective and its kernel has null square. Moreover, in some cases, this structure of affine bundle is passed down to Jet spaces. We give a characterization of this fact in algebraic terms. This algebraic condition also determines an affine structure between the groups of automorphisms of related Weil algebras