Cartan Connections on Lie Groupoids and their Integrability

Symmetry Integrability and Geometry-Methods and Applications

article

Anthony D. Blaom

关键词: Cartan connection;    Lie algebroid;    Lie groupoid;   

DOI  :  10.3842/SIGMA.2016.114

来源: National Academy of Science of Ukraine

【 摘 要 】

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 $G$.

【 授权许可】

Unknown