Inverting covariant exterior derivative

TL;DR

This paper presents an algorithm for inverting the covariant exterior derivative on small regions of fibered sets, connecting geometric methods with operator theory to solve related equations.

Contribution

It introduces a geometric, algorithmic approach to invert the covariant exterior derivative using the linear homotopy operator, with considerations on solution regularity.

Findings

The algorithm works on small star-shaped regions of fibered sets.

Constraints for invertibility are straightforward and relate to parallel transport equations.

The approach links geometric methods with operator theory for solving covariant equations.

Abstract

The algorithm for inverting covariant exterior derivative is provided. It works for a sufficiently small star-shaped region of a fibered set - a local subset of a vector bundle and associated vector bundle. The algorithm contains some constraints that can fail, giving no solution, which is the expected case for parallel transport equations. These constraints are straightforward to obtain in the proposed approach. The relation to operational calculus and operator theory is outlined. The upshot of this paper is to show, using the linear homotopy operator of the Poincare lemma, that we can solve the covariant constant and related equations in a geometric and algorithmic way. The considerations related to the regularity of the solutions are provided.