### Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

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 non-transitively 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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