= Holonomy
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= Holonomy of a connection
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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 $p h$ 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.
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