2024/09/19 by Čap, Andreas, Flood, Keegan J., Mettler, Thomas · 1 citation
#Differential Geometry (math.DG) #FOS: Mathematics #FOS: Physical sciences #Geometric Topology (math.GT) #Mathematical Physics (math-ph)
paper · doi:10.48550/arxiv.2409.12811
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 ℝ4 of a 3-manifold that is equipped with a torsion-free connection preserving a volume form.