nLab
holonomy group

Contents

Contents

Idea

For XX a space equipped with a GG-connection on a bundle \nabla (for some Lie group GG) and for xXx \in X any point, the parallel transport of \nabla assigns to each curve Γ:S 1X\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.

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.

Last revised on April 14, 2021 at 18:32:59. See the history of this page for a list of all contributions to it.