2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/162287We investigate the structure of conformal $C$-spaces,a class of Riemmanian manifolds which naturally arises as aconformal generalisation of the Einstein condition. A basic question is when such a structure is closed, or equivalently locally conformally Cotton. In dimension 4 we obtain a full answer to this question and also investigate the incidence of the Bach condition on this class of metrics. This is related to earlier results obtained in the Einstein-Weyl context.19 pagesDifferential Geometry53A30, 53C24, 53C55text