Weakly conformally symmetric manifolds
We study the properties of weakly conformally symmetric pseudo-Riemannian manifolds, with particular emphasis on the $4$-dimensional Lorentzian case. We provide a decomposition of the conformal curvature tensor in dimensions $n \geq 5$. Moreover, some 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: this notion was recently introduced and investigated by Mantica and Molinari. 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 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 shown that such a space-time is of Petrov type N with respect to the same vector.