Cartan Connections on Lie Groupoids and their Integrability

TL;DR

This paper explores Cartan connections on Lie groupoids, linking their integrability to the flatness of associated Cartan connections on Lie algebroids, and describes the structure of the groupoid of one-jets of bisections.

Contribution

It introduces a generalized notion of Cartan connections on Lie groupoids and establishes their integrability criteria via curvature conditions, extending classical differential geometry concepts.

Findings

A Cartan connection on a Lie groupoid induces a Cartan connection on its Lie algebroid.

Vanishing curvature of the infinitesimal connection characterizes the integrability of the geometric structure.

Provides a detailed description of the multiplication in the groupoid of one-jets of bisections.

Abstract

A multiplicatively closed, horizontal $n$-plane field $D$ on a Lie groupoid $G$ over $M$ generalizes to intransitive geometry the classical notion of a Cartan connection. The infinitesimalization of the connection $D$ is a Cartan connection $\nabla $ on the Lie algebroid of $G$, a notion already studied elsewhere by the author. It is shown that $\nabla $ may be regarded as infinitesimal parallel translation in the groupoid $G$ along $D$. From this follows a proof that $D$ defines a pseudoaction generating a pseudogroup of transformations on $M$ precisely when the curvature of $\nabla $ vanishes. A byproduct of this analysis is a detailed description of multiplication in the groupoid $J^1 G$ of one-jets of bisections of $G$.