In this thesis, Simons’ proof of Berger’s classification of nonsymmetric irreducible Riemannian manifolds with respect to their holonomy groups is studied and Berger’s classification is discussed. The main tools will be principal fibre bundles and vector bundles. Using them, the Ambrose-Singer theorem is investigated, which relates the geometric meaning of curvature to holonomy groups and forms the basis of Simons’ proof.
