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Inverting covariant exterior derivative

Radosław Antoni Kycia, Josef Šilhan

TL;DR

The paper develops explicit, local inversion formulas for the covariant exterior derivative $d^{\nabla}$ on star-shaped patches of fiber bundles, recasting the problem in terms of exact and antiexact decompositions via the linear homotopy operator $H$ and leveraging the Poincaré lemma to construct covariantly constant forms.Solutions are obtained for homogeneous and inhomogeneous equations, with precise convergence conditions (ball radius tied to $\|A\|_{\infty}$) and a clear handling of gauge/antiexact modes that affect uniqueness.The work extends to associated vector bundles, Hodge-dual settings, Cartan structure equations, curvature equations, and general geometric differential operators, and it situates the method within Bittner's operator calculus, offering an algorithmic, iterative framework for solving local differential-geometric problems.By providing a practical, geometric approach to inverting $d^{\nabla}$ and related operators, the paper furnishes a potentially impactful tool for gauge theory and differential-geometry computations, while also outlining regularity considerations and nonexistence constraints.

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.

Inverting covariant exterior derivative

TL;DR

The paper develops explicit, local inversion formulas for the covariant exterior derivative $d^{\nabla}$ on star-shaped patches of fiber bundles, recasting the problem in terms of exact and antiexact decompositions via the linear homotopy operator $H$ and leveraging the Poincaré lemma to construct covariantly constant forms.Solutions are obtained for homogeneous and inhomogeneous equations, with precise convergence conditions (ball radius tied to $\|A\|_{\infty}$) and a clear handling of gauge/antiexact modes that affect uniqueness.The work extends to associated vector bundles, Hodge-dual settings, Cartan structure equations, curvature equations, and general geometric differential operators, and it situates the method within Bittner's operator calculus, offering an algorithmic, iterative framework for solving local differential-geometric problems.By providing a practical, geometric approach to inverting $d^{\nabla}$ and related operators, the paper furnishes a potentially impactful tool for gauge theory and differential-geometry computations, while also outlining regularity considerations and nonexistence constraints.

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.
Paper Structure (21 sections, 22 theorems, 154 equations, 1 figure)

This paper contains 21 sections, 22 theorems, 154 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Inversion formula for covariant exterior derivative
  3. Homogenous equation
  4. Inhomogenous equation
  5. Exact inhomogenity
  6. General inhomogenity
  7. Associated vector bundles
  8. Horizontal projection
  9. Equivariance
  10. Relation to Bittner's operator calculus
  11. Curvature equation
  12. Hodge duals
  13. Cartan structure equations
  14. General geometric-based differential equations
  15. Conclusions
  16. ...and 6 more sections

Key Result

Proposition 1

The unique smooth solution $\phi\in \Lambda^{0,\infty}(U,\mathbb{R})$ to the equation is given by where $c \in \mathbb{R}$ is treated as an exact form (constant function) on $U$.

Figures (1)

  • Figure 1: Operator calculus mapped to exterior calculus.

Theorems & Definitions (60)

  • Definition 1
  • Proposition 1
  • Proof
  • Remark 1
  • Definition 2
  • Proposition 2
  • Proof
  • Corollary 1
  • Theorem 1
  • Proof
  • ...and 50 more