Ambrose–Singer curvature span

Let $\pi\colon P\to M$ be a principal $G$-bundle and let $\omega$ be a principal connection with curvature 2-form $\Omega\in \Omega^2(P;\mathfrak{g})$.

Fix $p\in P$.

Theorem (Ambrose–Singer)

Let $\mathfrak{hol}_p(\omega)\subset \mathfrak{g}$ be the holonomy algebra at $p$. Then $\mathfrak{hol}_p(\omega)$ is the smallest Lie subalgebra of $\mathfrak{g}$ containing all elements of the form

where:

• $q\in P$ can be reached from $p$ by horizontal transport along some piecewise smooth path in $M$,
• $X,Y\in T_qP$ are horizontal tangent vectors at $q$,
• $g\in G$ is the unique element such that horizontal transport along the chosen path sends $p$ to $q\cdot g$.

In words: the holonomy algebra is generated by curvature values computed at points reached by parallel transport , transported back to the basepoint by the adjoint action.

Examples

1. Curvature zero implies trivial holonomy algebra. If $\Omega\equiv 0$, then every generator above is zero, hence $\mathfrak{hol}_p(\omega)=0$.
2. 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 $\mathfrak{so}(n)$.
3. 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 $\mathfrak{h}$.