A groupoid approach to transitive differential geometry

TL;DR

This paper revisits classical differential geometry concepts using modern Lie groupoid and Morita equivalence techniques, introducing Cartan bundles and a unified flatness notion for various geometric structures.

Contribution

It develops a new framework encoding geometric structures via Cartan bundles and Morita equivalence, unifying classical integrability and flatness concepts.

Findings

Introduces Cartan bundles as principal G-bundles with transversally parallelisable foliations.

Defines a generalized flatness notion encompassing classical G-structure integrability.

Establishes Morita equivalence as a key tool in modern geometric structure analysis.

Abstract

This work is a spin-off of an on-going programme which aims at revisiting the original studies of Lie and Cartan on pseudogroups and geometric structures from a modern perspective. We encode geometric structures induced by transitive Lie pseudogroups into principal $G$-bundles equipped with a transversally parallelisable foliation generated by a subalgebra of $\mathfrak{g}$, called Cartan bundles. Our approach is complementary to arXiv:1911.13147 and is based on Morita equivalence of Lie groupoids. After identifying the main examples and properties, we develop a notion of flatness with respect to a Lie algebra, which encompasses the classical integrability of $G$-structures, the flatness of Cartan geometries, as well as the integrability of contact structures.