A Characterization of Invariant Connections

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 that 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.