holonomy group (changes) in nLab

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Idea

For XX a space equipped with a GG-connection on a bundle ∇\nabla (for some Lie group GG) and for x∈Xx \in X any point, the parallel transport of ∇\nabla assigns to each curve Γ:S 1→X\Gamma : S^1 \to X in XX starting and ending at xx an element hol ∇(γ)∈G hol_\nabla(\gamma) \in G: the holonomy of ∇\nabla along that curve.

The holonomy group of ∇\nabla at xx is the subgroup of GG on these elements.

If ∇\nabla is the Levi-Civita connection on a Riemannian manifold and the holonomy group is a proper subgroup HH of the special orthogonal group, one says that (X,g)(X,g) is a manifold of special holonomy .

Classification of holonomy groups of affine connections

Any closed Lie subgroup? of GL(V)GL(V) occurs as the holonomy group of some affine connection (with torsion, in general). See Hano–Ozeki [HanoOzeki].

Holonomy groups of locally symmetric connections can be classified using Élie Cartan’s classification of symmetric spaces?.

For Levi-Civita connections, holonomy groups were classified by Marcel Berger [Berger].

The case of torsion-free affine connections that are not locally symmetric and are not Levi-Civita connections was treated by Merkulov and Schwachhöfer [MerkulovSchwachhofer]. A complete list of exotic holonomy groups (for the metric and nonmetric cases) can be found in [MerkulovSchwachhofer2].

• connection on a bundle

  • connection on a 2-bundle, connection on an infinity-bundle

• parallel transport, higher parallel transport

• holonomy

  • holonomy group

  • special holonomy

References

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• J. Hano, H. Ozeki, On the holonomy groups of linear connections, Nagoya Math. J. 10, 97-100 (1956). doi.

• Marcel Berger, Sur les groupes d’holonomie homogènes de variétés à connexion affine et des variétés riemanniennes. Bulletin de la Société mathématique de France 79:null (1955), 279-330. doi.

• Sergei Merkulov, Lorenz Schwachhöfer. Classification of Irreducible Holonomies of Torsion-Free Affine Connections. Annals of Mathematics 150:1 (1999), 77–149. doi.

• Sergei Merkulov, Lorenz Schwachhöfer. Addendum to Classification of Irreducible Holonomies of Torsion-Free Affine Connections. Annals of Mathematics 150:3 (1999), 1177–1179. doi.