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A Characterization of Invariant Connections
Symmetry, integrability and geometry: methods and applications, Tome 10 (2014) Cet article a éte moissonné depuis la source Math-Net.Ru

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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.
Keywords: invariant connections; principal fibre bundles; loop quantum gravity; symmetry reduction. 
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     author = {Maximilian Hanusch},
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     url = {http://geodesic.mathdoc.fr/item/SIGMA_2014_10_a24/}
}
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Maximilian Hanusch. A Characterization of Invariant Connections. Symmetry, integrability and geometry: methods and applications, Tome 10 (2014). http://geodesic.mathdoc.fr/item/SIGMA_2014_10_a24/

[1] Ashtekar A., Bojowald M., Lewandowski J., “Mathematical structure of loop quantum cosmology”, Adv. Theor. Math. Phys., 7 (2003), 233–268, arXiv: gr-qc/0304074 | DOI | MR

[2] Bojowald M., Kastrup H. A., “Symmetry reduction for quantized diffeomorphism-invariant theories of connections”, Classical Quantum Gravity, 17 (2000), 3009–3043, arXiv: hep-th/9907042 | DOI | MR | Zbl

[3] Duistermaat J. J., Kolk J. A. C., Lie groups, Universitext, Springer-Verlag, Berlin, 2000 | DOI | MR | Zbl

[4] Fleischhack Ch., Loop quantization and symmetry: configuration spaces, arXiv: 1010.0449

[5] Hanusch M., Invariant connections in loop quantum gravity, arXiv: 1307.5303

[6] Harnad J., Shnider S., Vinet L., “Group actions on principal bundles and invariance conditions for gauge fields”, J. Math. Phys., 21 (1980), 2719–2724 | DOI | MR | Zbl

[7] Rudolph G., Schmidt M., Differential geometry and mathematical physics, v. I, Theoretical and Mathematical Physics, Manifolds, Lie groups and Hamiltonian systems, Springer, Dordrecht, 2013 | DOI | MR | Zbl

[8] Wang H. C., “On invariant connections over a principal fibre bundle”, Nagoya Math. J., 13 (1958), 1–19 | MR | Zbl

[9] Whitney H., “Differentiability of the remainder term in {T}aylor's formula”, Duke Math. J., 10 (1943), 153–158 | DOI | MR | Zbl

[10] Whitney H., “Differentiable even functions”, Duke Math. J., 10 (1943), 159–160 | DOI | MR | Zbl