Holonomy algebra - Knowlpedia

Let $\pi:P\to M$ be a principal $G$ -bundle with a principal connection $\omega$ , and choose $p\in P_x$ . Using the natural immersed Lie-group structure on the holonomy group (not its closure in $G$ ), the holonomy algebra at $p$ is

\mathfrak{hol}_p(\omega) :=\operatorname{Lie}\bigl(\operatorname{Hol}_p(\omega)\bigr) \subseteq\mathfrak g,

where $\operatorname{Hol}_p(\omega)\subseteq G$ is the holonomy group obtained by parallel transport around piecewise smooth loops based at $x$ , with horizontal lifts beginning at $p$ , and $\mathfrak g$ is the Lie algebra of $G$ .

Dependence on the basepoint

If $p$ is replaced by $p\cdot g$ in the same fiber, then $\operatorname{Hol}_{p\cdot g}(\omega)=g^{-1}\operatorname{Hol}_p(\omega)g$ . Thus the holonomy algebra changes by the corresponding adjoint action.

A key structural result is that $\mathfrak{hol}_p(\omega)$ is generated by curvature (see Ambrose–Singer curvature span).

Examples

Flat connections. If the curvature of $\omega$ vanishes, then $\mathfrak{hol}_p(\omega)=0$ .
Product structures. On a Riemannian product $M_1\times M_2$ with the product metric, the holonomy algebra splits as the direct sum of the holonomy algebras of the factors.