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.

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.