A fast convergence theorem for nearly multiplicative connections on proper Lie groupoids

Motivated by the study of a certain family of classical geometric problems we investigate the existence of multiplicative connections on proper Lie groupoids. We show that one can always deform a given connection which is only approximately multiplicative into a genuinely multiplicative connection. The proof of this fact that we present here relies on a recursive averaging technique. As an application we point out that the study of multiplicative connections on general proper Lie groupoids reduces to the study of longitudinal representations of regular groupoids. We regard our results as a preliminary step towards the elaboration of an obstruction theory for multiplicative connections.