Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 12 (2016), 114, 26 pages      arXiv:1605.04365      https://doi.org/10.3842/SIGMA.2016.114

Cartan Connections on Lie Groupoids and their Integrability

Anthony D. Blaom
10 Huruhi Road, Waiheke Island, New Zealand

Received May 19, 2016, in final form December 02, 2016; Published online December 07, 2016

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 $G$.

Key words: Cartan connection; Lie algebroid; Lie groupoid.

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