Holonomy and four-dimensional manifolds

This paper investigates the relationship between two fundamental types of objects associated with a connection on a manifold: the existence of parallel semi-Riemannian metrics and the associated holonomy group. Typically in Riemannian geometry, a metric is specified which determines a Levi-Civita connection. Here we consider the connection as more fundamental and allow for the possibility of several parallel metrics. Holonomy is an old geometric concept which is enjoying revived interest in certain branches of mathematical physics, in particular loop quantum gravity and Calabi-Yau manifolds in string theory. It measures, in group theoretic terms the connection’s deviation from flatness and takes the topology of the manifold into account. It is well known that for a Riemannian manifold, the reducibility of the holonomy group of the Levi-Civita connection implies the existence of multiple independent parallel Riemannian metrics on the manifold