Holonomy groups in Riemannian geometry

The curvature tensor generates the infinitesimal holonomy transformations. One can define a Surjective Group homomorphism from the Fundamental group of the manifold onto the quotient of the holonomy group by its connected component. The holonomy group detects the local re-ducibility of the manifold, and also whether the metric is locally symmetric.

In a certain sense, the classification theorem of Berger is the climax of these notes. It shows that in the riemannian context, all of the study that we have made of holonomy groups can be reduced to a short list of cases.