Natural operations on differential forms

TL;DR

This paper characterizes all natural operations on differential forms, showing they are limited to linear combinations, exterior products, and differentials, and applies this to classify forms associated with connections on principal bundles.

Contribution

It generalizes previous work by Palais and Freed-Hopkins, providing a comprehensive classification of natural operations on differential forms and their relation to principal bundle connections.

Findings

Only linear combinations, exterior products, and differentials are natural operations on differential forms.

Classifies natural differential forms associated with connections on principal bundles.

Extends previous classifications to a broader context.

Abstract

We prove that the only natural operations between differential forms are those obtained using linear combinations, the exterior product and the exterior differential. Our result generalises work by Palais and Freed-Hopkins. As an application, we also deduce a theorem, originally due to Kolar, that determines those natural differential forms that can be associated to a connection on a principal bundle.