The global geometry of Riemannian manifolds with commuting curvature operators

We give manifolds whose Riemann curvature operators commute, i.e. which satisfy for all tangent vectors x_i in both the Riemannian and the higher signature settings. 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. Dedicated to the memory of Jean Leray