 Holonomy

Let $P\to B$ be a principal bundle with principal connection. Fix a point $b\in B$ and a point $p\in P_b$. The horizontal lift of a piecewise smooth closed curve $\gamma$ based at $b$ starts at $p$ and ends at a unique $p{g}_{\gamma }$. This group element $g_{\gamma }$ is its holonomy relative to $p$.

Changing the starting point to $ph$ changes the group element to $h^{-1}{g}_{\gamma }h$, by equivariance of parallel transport. The conjugacy class is therefore independent of the chosen fiber frame. Transport along a path to another basepoint similarly conjugates the resulting group of holonomy elements.

For a flat principal connection, contractible closed curves have trivial holonomy, but noncontractible closed curves can return a different fiber frame. For example, the U(1) connection $A=\alpha d\theta$ on a trivial circle bundle over a circle has zero curvature form and holonomy $e^{2\pi i\alpha }$ around one circuit. This separates a local curvature condition from a global transport obstruction.

Holonomy is a concept from differential geometry and mathematical physics that describes the behavior of parallel transport around closed loops in a manifold. It provides insight into the geometric properties of the space, including curvature and how certain geometric structures behave under parallel transport.