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.
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The holonomy of a connection along a closed curve is the group element relating the endpoints of its horizontal lift. A flat connection can have nontrivial holonomy around a noncontractible loop.