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

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.

💡 Research Summary

The paper introduces the concept of a flat extension of a connection θ on a principal G‑bundle. Given a Lie group ˜G containing G as a subgroup and equipped with an Ad‑invariant symmetric bilinear form ⟨·,·⟩ on its Lie algebra ˜𝔤, one can decompose the Maurer–Cartan form µ˜G into a 𝔤‑valued part µ˜G^⊤ and an orthogonal complement µ˜G^⊥. A flat extension of type (˜G,G) is a bundle map F : P → ˜G such that θ = F*µ˜G^⊤. The central algebraic result (Theorem 5.1) shows that for any ˜𝔤‑valued 1‑form ψ satisfying the Maurer–Cartan equation dψ + ½