The integration problem for principal connections
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866911019873337344 |
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| author | Fernandez, Javier Kordon, Francisco |
| author_facet | Fernandez, Javier Kordon, Francisco |
| contents | In this paper we introduce the Integration Problem for principal connections. Just as a principal connection on a principal bundle $ϕ:Q\rightarrow M$ may be used to split $TQ$ into horizontal and vertical subbundles, a discrete connection may be used to split $Q\times Q$ into horizontal and vertical submanifolds. All discrete connections induce a connection on the same principal bundle via a process known as the Lie or derivative functor. The Integration Problem consists of describing, for a principal connection $\mathcal{A}$, the set of all discrete connections whose associated connection is $\mathcal{A}$. Our first result is that for \emph{flat} principal connections, the Integration Problem has a unique solution among the \emph{flat} discrete connections. More broadly, under a fairly mild condition on the structure group $G$ of the principal bundle $ϕ$, we prove that the existence part of the Integration Problem has a solution that needs not be unique. Last, we see that, when $G$ is abelian, given compatible continuous and discrete curvatures the Integration Problem has a unique solution constrained by those curvatures. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2407_13614 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | The integration problem for principal connections Fernandez, Javier Kordon, Francisco Differential Geometry Mathematical Physics Dynamical Systems 53B15, 53C05 (Primary) 37J06, 70G45 (Secondary) In this paper we introduce the Integration Problem for principal connections. Just as a principal connection on a principal bundle $ϕ:Q\rightarrow M$ may be used to split $TQ$ into horizontal and vertical subbundles, a discrete connection may be used to split $Q\times Q$ into horizontal and vertical submanifolds. All discrete connections induce a connection on the same principal bundle via a process known as the Lie or derivative functor. The Integration Problem consists of describing, for a principal connection $\mathcal{A}$, the set of all discrete connections whose associated connection is $\mathcal{A}$. Our first result is that for \emph{flat} principal connections, the Integration Problem has a unique solution among the \emph{flat} discrete connections. More broadly, under a fairly mild condition on the structure group $G$ of the principal bundle $ϕ$, we prove that the existence part of the Integration Problem has a solution that needs not be unique. Last, we see that, when $G$ is abelian, given compatible continuous and discrete curvatures the Integration Problem has a unique solution constrained by those curvatures. |
| title | The integration problem for principal connections |
| topic | Differential Geometry Mathematical Physics Dynamical Systems 53B15, 53C05 (Primary) 37J06, 70G45 (Secondary) |
| url | https://arxiv.org/abs/2407.13614 |