On the de Rham–Wu decomposition for Riemannian and Lorentzian manifolds

Abstract

It is explained how to find the de Rham decomposition of a Riemannian manifold and the Wu decomposition of a Lorentzian manifold. For that it is enough to find parallel symmetric bilinear forms on the manifold, and do some linear algebra. This result will allow to compute the connected holonomy group of an arbitrary Riemannian or Lorentzian manifold.