Ambrose–Singer curvature span

Let $\pi\colon P\to M$ be a principal $G$ -bundle with principal connection $\omega$ and curvature $\Omega\in\Omega^2(P;\mathfrak g)$ . Fix $p\in P$ . The Ambrose–Singer theorem states that the holonomy algebra $\mathfrak{hol}_p(\omega)\subseteq\mathfrak g$ is the Lie subalgebra spanned by

\Omega_q(X,Y),

where $q\in P$ ranges over points reachable from $p$ by horizontal lifts of piecewise smooth paths in $M$ , and $X,Y\in T_qP$ are horizontal tangent vectors.

Remarks

Curvature values at horizontally reachable points already use the frames obtained from $p$ by parallel transport. An equivalent formulation computes curvature in arbitrary frames and transports the resulting elements of $\mathfrak g$ back by the adjoint action.

Examples

Curvature zero implies trivial holonomy algebra. If $\Omega\equiv 0$ , then every generator above is zero, hence $\mathfrak{hol}_p(\omega)=0$ .
Constant-curvature Riemannian metrics. For the Levi–Civita connection of a round sphere, the curvature endomorphisms span $\mathfrak{so}(n)$ , so the holonomy algebra is all of $\mathfrak{so}(n)$ .
Reduced structure group. If a connection reduces to a subgroup $H\subset G$ (so its connection form takes values in $\mathfrak{h}$ ), then every curvature value lies in $\mathfrak{h}$ and the holonomy algebra is contained in $\mathfrak{h}$ .