On weakly conformally symmetric pseudo-Riemannian manifolds

In this paper, we study the properties of weakly conformally symmetric pseudo-Riemannian manifolds focusing particularly on the 4-dimensional Lorentzian case. First, we provide a new proof of an important result found in literature; then several new others are stated. We provide a decomposition for the conformal curvature tensor in n >= 5. Moreover, some important identities involving two particular covectors are stated; for example, it is proven that under certain conditions the Ricci tensor and other tensors are Weyl compatible. Topological properties involving the vanishing of the first Pontryagin form are then stated. Further, we study weakly conformally symmetric 4-dimensional Lorentzian manifolds (space-times): it is proven that one of the previously defined covectors is null and unique up to a scaling. Moreover, it is shown that under certain conditions, the same vector is an eigenvector of the Ricci tensor and its integral curves are geodesics. Finally, it is stated that such space-time is of Petrov type N with respect to the same vector.