Holonomy group - Knowlpedia

Let $\pi:P\to M$ be a principal $G$ -bundle equipped with a principal connection. Fix a point $p\in P$ and set $x=\pi(p)\in M$ .

For a piecewise smooth loop $\gamma:[0,1]\to M$ with $\gamma(0)=\gamma(1)=x$ , let $\widetilde\gamma:[0,1]\to P$ be the horizontal lift starting at $p$ (i.e. $\widetilde\gamma(0)=p$ and $\dot{\widetilde\gamma}(t)\in H_{\widetilde\gamma(t)}$ on each smooth piece). Since $\widetilde\gamma(1)$ lies in the same fiber as $p$ , there is a unique $g_\gamma\in G$ such that

\widetilde\gamma(1)=p\cdot g_\gamma.

The holonomy group at $p$ is

\mathrm{Hol}_p \coloneqq \{\, g_\gamma \in G \mid \gamma \text{ is a loop based at } x \,\}\;\subset\; G.

If $p' = p\cdot h$ is another point in the same fiber, then $\mathrm{Hol}_{p'} = h^{-1}\mathrm{Hol}_p\,h$ ; thus the holonomy group is well-defined up to conjugacy inside $G$ .

Lie-group structure

The holonomy group has a natural immersed Lie-group structure. It need not be closed in $G$ ; this Lie-group structure must not be replaced by the topology of its closure in $G$ . Its identity component in this structure is the restricted holonomy group.

Equivalent characterizations

Equivalently, $\mathrm{Hol}_p$ is the subgroup generated by all parallel transports along loops based at $x$ .

Examples

Trivial flat connection. On $M\times G$ with the flat product connection, horizontal lifts keep the $G$ -coordinate constant, so $\mathrm{Hol}_p=\{e\}$ .

Flat connection over $S^1$ . For a flat connection on a bundle over $S^1$ , parallel transport around the fundamental loop yields an element $h\in G$ . The holonomy group is the cyclic subgroup generated by $h$ , which need not be closed in $G$ .

Hopf fibration. For the standard connection on the Hopf bundle $S^3\to S^2$ with structure group $U(1)$ , holonomy around loops in $S^2$ produces arbitrary phases, so $\mathrm{Hol}_p = U(1)$ .