Cartan Connections on Lie Groupoids and their Integrability

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 $G$.

References in corpus (3)

• Lectures on Integrability of Lie Brackets
• Riemannian metrics on differentiable stacks
• Multiplicative Forms and Spencer Operators

Cited by in corpus (4)

• Cartan geometries and multiplicative forms
• Integration of quadratic Lie algebroids to Riemannian Cartan-Lie groupoids
• Lie Algebroid Invariants for Subgeometry
• The Lie algebroid associated with a hypersurface