Multiplicative Ehresmann connections

TL;DR

This paper develops the theory of multiplicative Ehresmann connections for Lie groupoids and algebroids, establishing existence, obstructions, and analogues of classical connection concepts, with applications to Poisson geometry.

Contribution

It introduces the theory of multiplicative Ehresmann connections, including existence criteria, obstructions, and their classical analogues in a Lie groupoid context.

Findings

Constructed obstructions to existence of multiplicative Ehresmann connections.

Proved existence for all proper Lie groupoids and several classes.

Applied results to Poisson geometry and linearization around Poisson submanifolds.

Abstract

We develop the theory of multiplicative Ehresmann connections for Lie groupoid submersions covering the identity, as well as their infinitesimal counterparts. We construct obstructions to the existence of such connections, and we prove existence for several interesting classes of Lie groupoids and Lie algebroids, including all proper Lie groupoids. We show that many notions from the theory of principal bundle connections have analogues in this general setup, including connections 1-forms, curvature 2-forms, Bianchi identity, etc. In [17] we provide a non-trivial application of the results obtained here to construct local models in Poisson geometry and to obtain linearization results around Poisson submanifolds.