Cartan connections on Lie algebroids and their integrability

Anthony D. Blaom

Published 2016-05-14Version 1

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 $G$.