Abstract Let Mᵐ M m be a complete, connected, noncompact m -dimensional Riemannian manifold and ξ be a nontrivial closed conformal vector field on M, with at least one singular point, say p, and conformal factor ψ. We show that, when m>2 m > 2 or when m=2 m = 2 and the singular set of ξ consists of isolated points at which ψ does not vanish, then p is the only singular point of ξ and ₚ: TₚM M exp p: T p M → M is a diffeomorphism. Then, we use this fact to present a formula, built on | | | ξ |, for the Riemannian volume of geodesic balls of M centered at p. When m=2 m = 2, such a formula generates necessary and sufficient conditions for M to be: (i) conformally equivalent to the Euclidean or hyperbolic plane; (ii) of finite total curvature. Finally, after showing that the conformal factor can be prescribed under some conditions, we finish the paper proving that Cᵐ C m is the only example in the class of Kähler manifolds of complex dimension m>1 m > 1.
Caminha et al. (Fri,) studied this question.