Holonomy

Summary

The holonomy of a connection records how parallel transport around closed loops transforms the fibre at a base point. These transformations form a group, the holonomy group, which is a Lie subgroup of the general linear group of the fibre. The holonomy group measures curvature globally: it is trivial for flat connections on simply connected manifolds. For Riemannian manifolds the possible holonomy groups are classified, and special holonomy groups correspond to special geometries such as Kähler, Calabi–Yau and $G_2$ manifolds.