Multiplicative Connections and Their Lie Theory

TL;DR

This paper introduces and studies multiplicative connections on Lie groupoids, exploring their properties, obstructions, and infinitesimal counterparts, with examples illustrating the concepts.

Contribution

It defines multiplicative connections on Lie groupoids, characterizes their properties, and establishes an integration theorem for their infinitesimal versions.

Findings

Characterization of multiplicative connections via torsion and geodesic spray

Identification of obstructions to existence of such connections

Establishment of an integration theorem for IM connections

Abstract

We define and study multiplicative connections in the tangent bundle of a Lie groupoid. Multiplicative connections are linear connections satisfying an appropriate compatibility with the groupoid structure. Our definition is natural in the sense that a linear connection on a Lie groupoid is multiplicative if and only if its torsion is a multiplicative tensor in the sense of Bursztyn-Drummond [5] and its geodesic spray is a multiplicative vector field. We identify the obstruction to the existence of a multiplicative connection. We also discuss the infinitesimal version of multiplicative connections in the tangent bundle, that we call infinitesimally multiplicative (IM) connections and we prove an integration theorem for IM connections. Finally, we present a few toy examples.