Flat extensions of principal connections and the Chern-Simons $3$-form
Fuente:
arXiv
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| Autori principali: | , , |
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| Natura: | Preprint |
| Pubblicazione: |
2024
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| _version_ | 1866915816280162304 |
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| author | Čap, Andreas Flood, Keegan J. Mettler, Thomas |
| author_facet | Čap, Andreas Flood, Keegan J. Mettler, Thomas |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2409_12811 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Flat extensions of principal connections and the Chern-Simons $3$-form Čap, Andreas Flood, Keegan J. Mettler, Thomas Differential Geometry Mathematical Physics Geometric Topology 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. |
| title | Flat extensions of principal connections and the Chern-Simons $3$-form |
| topic | Differential Geometry Mathematical Physics Geometric Topology |
| url | https://arxiv.org/abs/2409.12811 |