nLab holonomy group

Contents

Context

Differential cohomology

differential cohomology

Contents

Idea

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

The holonomy group of $\nabla$ at $x$ is the subgroup of $G$ on these elements.

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

Classification of holonomy groups of affine connections

Any closed Lie subgroup? of $GL(V)$ 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.

References

• 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.

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