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Darboux-Lie derivatives

Antonio De Nicola, Ivan Yudin

TL;DR

This work develops a unified calculus for derivatives of fiber-bundle maps called the Darboux-Lie derivative, defined for maps from natural bundles to associated bundles via a Trautman lift and $G$-equivariant data. It shows that Lie and covariant derivatives are special cases and proves key structural results: flow characterizations, Leibniz-type rules for products, tensor products, and compositions, and a Cartan-type magic formula in the covariant setting. The framework is designed to advance the theory of $G$-structures by providing tools to study gauge equivalence classes of soldering forms and their automorphisms. The results establish a robust, coordinate-free calculus with potential applications in differential geometry and gauge theory related to $G$-structures.

Abstract

We introduce the Darboux-Lie derivative for fiber-bundle maps from natural bundles to associated fiber bundles and study its properties.

Darboux-Lie derivatives

TL;DR

This work develops a unified calculus for derivatives of fiber-bundle maps called the Darboux-Lie derivative, defined for maps from natural bundles to associated bundles via a Trautman lift and -equivariant data. It shows that Lie and covariant derivatives are special cases and proves key structural results: flow characterizations, Leibniz-type rules for products, tensor products, and compositions, and a Cartan-type magic formula in the covariant setting. The framework is designed to advance the theory of -structures by providing tools to study gauge equivalence classes of soldering forms and their automorphisms. The results establish a robust, coordinate-free calculus with potential applications in differential geometry and gauge theory related to -structures.

Abstract

We introduce the Darboux-Lie derivative for fiber-bundle maps from natural bundles to associated fiber bundles and study its properties.
Paper Structure (19 sections, 9 theorems, 127 equations)

This paper contains 19 sections, 9 theorems, 127 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Principal bundles
  4. Associated bundles
  5. Vector fields and their flows
  6. Natural bundles
  7. Derivatives
  8. Darboux-Lie derivatives
  9. Examples and relation to other definitions
  10. Lie derivative
  11. Covariant derivative
  12. Darboux-Lie derivatives with respect to vertical right-invariant vector fields
  13. Leibniz rule
  14. Darboux-Lie derivative of a product
  15. Darboux-Lie derivative of a tensor product
  16. ...and 4 more sections

Key Result

Proposition 2.1

Let $(F,p)$ be a natural bundle on $\mathcal{M}f_n$. For every $M \in \mathcal{M}f_n$ there is a unique well-defined map such that for each $X \in \mathfrak{X}(M)$

Theorems & Definitions (29)

  • Proposition 2.1
  • proof
  • Remark 3.1
  • Definition 3.2
  • Remark 3.3
  • Example 3.4: Lie derivative
  • Example 3.5: Covariant derivative
  • Proposition 3.6
  • Corollary 3.7
  • Definition 4.1
  • ...and 19 more