On the naturality of the exterior differential

TL;DR

This paper investigates conditions under which the exterior differential commutes with Sobolev mappings, ensuring the naturality of the exterior differential operator in a Sobolev space context.

Contribution

It provides sufficient conditions for the naturality of the exterior differential under Sobolev mappings, extending classical results to less regular maps.

Findings

Established criteria for the naturality of the exterior differential with Sobolev maps

Extended the validity of the exterior differential equation to Sobolev settings

Provided mathematical conditions ensuring the commutation of exterior differential and pullback operations

Abstract

We give sufficient conditions for the naturallity of the exterior differential under Sobolev mappings. In other words we study the validity of the equation $d f^* \alpha = f^* d\alpha$ for a smooth form $\alpha$ and a Sobolev map $f$.