Lie algebroids and Cartan's method of equivalence

TL;DR

This paper reformulates Cartan's equivalence method using Lie algebroids, providing a coordinate-free geometric framework that enables the construction of normal forms and the interpretation of symmetries and curvature in geometric structures.

Contribution

It introduces a Lie algebroid-based formalism for Cartan's equivalence method, allowing for a full geometric interpretation and the construction of normal forms for finite-type objects.

Findings

Developed a coordinate-free geometric formalism for Cartan's method

Constructed Cartan algebroids as normal forms for finite-type objects

Applied the framework to subriemannian contact structures and conformal geometry

Abstract

Elie Cartan's general equivalence problem is recast in the language of Lie algebroids. The resulting formalism, being coordinate and model-free, allows for a full geometric interpretation of Cartan's method of equivalence via reduction and prolongation. We show how to construct certain normal forms (Cartan algebroids) for objects of finite-type, and are able to interpret these directly as infinitesimal symmetries deformed by curvature.' Details are developed for transitive structures but rudiments of the theory include intransitive structures (intransitive symmetry deformations). Detailed illustrations include subriemannian contact structures and conformal geometry.