The holonomy group of a connection on a manifold measures the failure of parallel transport to be path-independent: transporting a vector around a closed loop based at a point returns it acted on by an element of the structure group, and the set of all such transformations, ranging over all loops at , forms a group . For the Levi-Civita connection of a Riemannian metric parallel transport preserves the inner product, so , and on an oriented manifold . The groups at different base points are conjugate, so on a connected manifold one speaks of the holonomy group up to conjugacy.
Berger’s list classifies the possible holonomy groups of a simply connected, irreducible, non-symmetric Riemannian manifold: (generic), , , , , , and . The reduced-holonomy entries each impose a parallel geometric structure. Holonomy is equivalent to the manifold being Kähler. Holonomy gives a Ricci-flat Kähler manifold, i.e. a Calabi–Yau manifold. Holonomy gives a hyperkähler manifold. The cases (in dimension ) and (in dimension ) are the two exceptional holonomies. In particular ” holonomy” is exactly the condition defining a Calabi–Yau threefold.