Rough geometric integration

TL;DR

This paper introduces a new framework for integrating distributional k-forms on manifolds, combining geometric and sewing approaches to rough integration, enabling integration against regular submanifolds.

Contribution

It presents a novel notion of distributional k-forms on manifolds that unifies geometric and sewing methods for rough integration.

Findings

Defines distributional k-forms compatible with rough integration

Combines Whitney's geometric integration with sewing approaches

Enables integration against regular submanifolds

Abstract

We introduce a notion of distributional $k$-forms on $d$-dimensional manifolds which can be integrated against suitably regular $k$-submanifolds. Our approach combines ideas from Whitney's geometric integration [Whi57] with those of sewing approaches to rough integration [Gub04, FdLP06].