  In this paper we investigate the relationship between the existence of
parallel semi-Riemannian metrics of a connection and the reducibility of the
associated holonomy group. The question as to whether the holonomy group
necessarily reduces in the presence of a specified number of independent
parallel semi-Riemannian metrics is completely determined by the the signature
of the metrics and the dimension, d, of the manifold, when d not equal to four.
In particular, the existence of two independent, parallel semi-Riemannian
metrics, one of which having signature (p,q) with p not equal to q, implies the
holonomy group is reducible. The (p,p) cases, however, may allow for more than
one parallel metric and yet an irreducible holonomy group: for n=2m, m >2,
there exist connections on n-dimensional Euclidean space with irreducible
infinitesimal holonomy and which have two independent, parallel metrics of
signature (m,m). The case of four-dimensional manifolds, however, depends on
the topology of the manifold in question: the presence of three parallel
metrics always implies reducibility but reducibility in the case of two metrics
of signature (2,2) is guaranteed only for simply connected manifolds. The main
theorem in the paper is the construction of a topologically non-trivial
four-dimensional manifold with a connection that admits two independent
parallel metrics of signature (2,2) and yet has irreducible holonomy. We
provide a complete solution to the general problem.
