Volume 8 (2025)
Čap, Andreas;  Flood, Keegan J.;  Mettler, Thomas
Flat extensions of principal connections and the Chern–Simons 3-form
Annales Henri Lebesgue, Volume 8 (2025), pp. 1037-1059

Metadata

DOI10.5802/ahl.255
Keywords Chern–Simons invariants ,  principal connections ,  flat extensions

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.


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