Closed conformal vector fields on pseudo-Riemannian manifolds

∇XV = λX for every vector field X. (1.2) Here ∇ denotes the Levi-Civita connection of g. We call vector fields satisfying (1.2) closed conformal vector fields. They appear in the work of Fialkow [3] about conformal geodesics, in the works of Yano [7–11] about concircular geometry in Riemannian manifolds, and in the works of Tashiro [6], Kerbrat [4], Kühnel and Rademacher [5], and many other authors. If V is lightlike on (M,g), then from (1.2), we get Xg(V ,V)= 2g∇XV ,V = 2λg(X ,V)= 0 (1.3) for every vector field X . Thus λ≡ 0 and V is parallel. About lightlike parallel vector fields, we have the following theorem.