
SIGMA 10 (2014), 025, 25 pages arXiv:1310.0318
https://doi.org/10.3842/SIGMA.2014.025
A Characterization of Invariant Connections
Maximilian Hanusch
Department of Mathematics, University of Paderborn, Warburger Straße 100, 33098 Paderborn, Germany
Received December 09, 2013, in final form March 10, 2014; Published online March 15, 2014;
Example 4.10.2 and Lemma 3.7.2 revised, statement added to Remark 3.6.3, misprints corrected January 27, 2015
Abstract
Given a principal fibre bundle with structure group $S$, and a fibre transitive Lie group $G$ of automorphisms
thereon, Wang's theorem identifies the invariant connections with certain linear maps $\psi\colon
\mathfrak{g}\rightarrow \mathfrak{s}$.
In the present paper, we prove an extension of this theorem which applies to the general situation where $G$ acts
nontransitively on the base manifold.
We consider several special cases of the general theorem, including the result of Harnad, Shnider and Vinet which applies
to the situation where $G$ admits only one orbit type.
Along the way, we give applications to loop quantum gravity.
Key words:
invariant connections; principal fibre bundles; loop quantum gravity; symmetry reduction.
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