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

TL;DR

This paper presents a method to find the de Rham and Wu decompositions of Riemannian and Lorentzian manifolds by identifying parallel symmetric bilinear forms, enabling computation of their holonomy groups.

Contribution

It introduces a linear algebra-based approach to determine decompositions and holonomy groups for Riemannian and Lorentzian manifolds, simplifying previous methods.

Findings

Method to find parallel symmetric bilinear forms

Procedure to compute holonomy groups

Application to 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.