Groups of Curvature Collineations in Riemannian Space-Times Which Admit Fields of Parallel Vectors

Abstract

By definition, a Riemannian space Vn admits a symmetry called a curvature collineation (CC) if the Lie derivative with respect to some vector ξi of the Riemann curvature tensor vanishes. It is shown that if a Vn admits a parallel vector field, then it will admit groups of CC's. It follows that every space-time with an expansion-free, shear-free, rotation-free, geodesic congruence admits groups of CC's, and hence gravitational pp waves admit such groups of symmetries.