nLab
holonomy group

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 ${\mathrm{hol}}_{\nabla }(\gamma )\in G$: 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 .

Related concepts

• 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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