Riemannian Holonomy Groups

Abstract

Abstract Let M be a manifold, and g a Riemannian metric on M . Then there is a unique, preferred connection on TM called the Levi-Civita connection, which is torsion-free and satisfies g = 0. The curvature R( ) of the Levi-Civita connection is called the Riemann curvature, and its holonomy group Hol( ) the Riemannian holonomy group Hol( g) of g. In 1955, Marcel Berger proved that if ( M, g) is a Riemannian manifold with M simply-connected and g irreducible and nonsymmetric, then Hol( g) must be one of SO( n), U( m), SU( m), Sp( m), Sp( m) Sp(1), G 2 or Spin(7). The goal of 3.1– 3.4 is to explain what this result means and how it is proved. We start with the Levi-Civita connection, Riemann curvature, and Riemannian holonomy groups. After sections on reducible Riemannian manifolds and symmetric spaces we move onto Berger ‘s classification, describing the proof and the groups on Berger ‘s list. Sections 3.5 and 3.6 explore the relationship between the holonomy group Hol( g) and the topology of the underlying manifold M , in particular when M is compact.