Flat extensions of principal connections and the Chern-Simons $3$-form

TL;DR

This paper explores flat extensions of principal connections, linking their existence to the vanishing of Chern-Simons invariants, and applies these ideas to geometric embedding problems of 3-manifolds.

Contribution

It introduces the concept of flat extensions of principal connections and relates them to Chern-Simons invariants, providing new obstructions for geometric immersions.

Findings

Flat extensions correspond to pull-backs of Maurer-Cartan forms.

Vanishing Chern-Simons invariant indicates existence of certain flat extensions.

Obstructions for conformal and equiaffine immersions of 3-manifolds are established.

Abstract

We introduce the notion of a flat extension of a connection $\theta$ on a principal bundle. Roughly speaking, $\theta$ admits a flat extension if it arises as the pull-back of a component of a Maurer-Cartan form. For trivial bundles over closed oriented $3$-manifolds, we relate the existence of certain flat extensions to the vanishing of the Chern-Simons invariant associated with $\theta$. As an application, we recover the obstruction of Chern-Simons for the existence of a conformal immersion of a Riemannian $3$-manifold into Euclidean $4$-space. In addition, we obtain corresponding statements for a Lorentzian $3$-manifold, as well as a global obstruction for the existence of an equiaffine immersion into $\mathbb{R}^4$ of a $3$-manifold that is equipped with a torsion-free connection preserving a volume form.