Affine bundles are affine spaces over modules

TL;DR

This paper establishes an equivalence between affine bundles over smooth manifolds and affine spaces modeled on projective finitely generated modules, providing new insights into the structure of vector bundles and their differential operators.

Contribution

It introduces an equivalence of categories linking affine bundles to modules, offering an alternative proof for the characterization of vector bundles via differential operators.

Findings

Category of affine bundles is equivalent to affine spaces over modules

Characterization of vector bundles extends to rank 1 bundles over any base manifold

Provides an alternative proof for a known main result

Abstract

We show that the category of affine bundles over a smooth manifold M is equivalent to the category of affine spaces modelled on projective finitely generated C^\infty(M)-modules. Using this equivalence of categories, we are able to give an alternate proof of the main result of [13], showing that the characterization of vector bundles by means of their Lie algebras of homogeneous differential operators also holds for vector bundles of rank 1 and over any base manifolds.