Holonomy algebra

Let $\pi\colon P\to M$ be a principal G-bundle with structure group $G$, and let $\omega$ be a principal connection on $P$.

Choose a basepoint $x\in M$ and a point $p\in P_x$ in the fiber.

Definition (Holonomy algebra)

The holonomy group $\mathrm{Hol}_p(\omega)\subset G$ is the subgroup consisting of all elements obtained by parallel transport along piecewise smooth loops in $M$ based at $x$, lifted horizontally starting at $p$.

The holonomy algebra at $p$ is the Lie algebra

where $\mathfrak{g}$ is the Lie algebra of $G$.

If $p$ is replaced by $p\cdot g$ in the same fiber, then $\mathrm{Hol}_{p\cdot g}(\omega)=g^{-1}\mathrm{Hol}_p(\omega)g$, so $\mathfrak{hol}_p(\omega)$ is well-defined up to conjugacy in $\mathfrak{g}$.

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

Examples

1. Flat connections. If the curvature of $\omega$ vanishes, then $\mathfrak{hol}_p(\omega)=0$ (and the holonomy group is locally constant).
2. Generic Riemannian holonomy. For a generic Riemannian metric in dimension $n$, the Levi–Civita connection has holonomy algebra $\mathfrak{so}(n)$.
3. Product structures. On a Riemannian product $M_1\times M_2$ with product metric, the holonomy algebra splits as a direct sum corresponding to the factors, reflecting the decomposition of parallel transport.