Flat extensions of principal connections and the Chern-Simons $3$-form
Andreas Čap, Keegan J. Flood, Thomas Mettler
TL;DR
The paper develops a framework linking flat extensions of principal connections to Chern–Simons invariants on 3-manifolds. By embedding a Lie algebra 𝔤 into a larger Ṽ𝔤 and exploiting symmetric pairs, it shows that flat extensions force the CS invariant to vanish or become integral, yielding obstructions to geometric immersions. The authors instantiate the theory in Riemannian, Lorentzian, and equiaffine contexts, deriving concrete obstructions for conformal immersions into ℝ^4, isometric immersions into Lorentzian spaces, and equiaffine immersions of 3-manifolds with volume-preserving connections. The approach leverages a key partial-blindness identity for CS forms and provides explicit normalizations ensuring integrality results, with applications including RP^3 and CR-geometry examples.
Abstract
We introduce the notion of a flat extension of a connection $θ$ on a principal bundle. Roughly speaking, $θ$ 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 $θ$. 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.