Restricted holonomy group - Imported Knowlpedia Content

Let $\pi:P\to M$ be a principal $G$ -bundle with connection, and let $\mathrm{Hol}_p$ be the holonomy group at $p\in P$ .

The restricted holonomy group at $p$ , denoted $\mathrm{Hol}_p^0$ , is the subgroup of $\mathrm{Hol}_p$ generated by parallel transport around loops $\gamma$ based at $x=\pi(p)$ that are contractible in $M$ .

If $M$ is a connected smooth manifold, then $\mathrm{Hol}_p^0$ is a connected Lie subgroup of $G$ , and it coincides with the identity component of $\mathrm{Hol}_p$ . As with full holonomy, changing $p$ within the fiber conjugates $\mathrm{Hol}_p^0$ .

Examples

Simply connected base. If $M$ is simply connected, every loop is contractible, hence $\mathrm{Hol}_p^0=\mathrm{Hol}_p$ .

Flat connections. For a flat principal connection, holonomy depends only on the homotopy class of the loop. Contractible loops represent the identity element of $\pi_1(M)$ , so $\mathrm{Hol}_p^0=\{e\}$ , while $\mathrm{Hol}_p$ may still be nontrivial if $\pi_1(M)$ is nontrivial.

Discrete structure group. If $G$ is discrete (e.g. $G=O(1)=\{\pm1\}$ ), then the identity component of any subgroup is trivial. Thus $\mathrm{Hol}_p^0=\{e\}$ always, even when $\mathrm{Hol}_p$ is nontrivial.