  We give manifolds in both the Riemannian and in the higher signature settings
whose Riemann curvature operators commute, i.e. which satisfy
R(a,b)R(c,d)=R(c,d)R(a,b) for all tangent vectors. These manifolds have global
geometric phenomena which are quite different for higher signature manifolds
than they are for Riemannian manifolds. Our focus is on global properties;
questions of geodesic completeness and the behaviour of the exponential map are
investigated.
