The first part of the paper contains a well-written short survey of the theory of conformal transformations and conformal vector fields on pseudo-Riemannian manifolds, including recent results of the authors on conformally gradient vector fields. Generalizing these results, the authors prove the existence of the infinite-dimensional space of non-conformally flat pseudo-Riemannian manifolds admitting a complete conformal gradient vector field with at least one zero. They also study a conformally closed conformal vector field \(X\) in a pseudo-Riemannian manifold \((M,g)\), that is, a conformal field such that the 1-form \(e^fg\circ X\) is closed for a certain function \(f\). The following theorem is proved.
Theorem. Let \(M\) be a pseudo-Riemannian manifold of signature \((p,q)\) admitting a conformally closed conformal vector field \(X\) with at least one zero. Then:
(i) In a neighbourhood of the zero, the manifold is conformally flat.
(ii) If \(X\) is complete, then there is a globally defined conformal immersion of the pseudo-Euclidean space \(E^{p,q}\) into \(M\).
(iii) If, in addition, \(M\) is compact, then the metric must be positive or negative definite and \(M\) is conformally equivalent to the standard sphere.