On Closed Conformal Vector Fields with Singularities

Let Mᵐ M m be a complete, connected, noncompact m -dimensional Riemannian manifold and ξ ξ be a nontrivial closed conformal vector field on M , with at least one singular point, say p , and conformal factor ψ ψ . We show that, when $$m>2$$ m > 2 or when $$m=2$$ m = 2 and the singular set of ξ ξ consists of isolated points at which ψ ψ does not vanish, then p is the only singular point of ξ ξ and exp ₚ:TₚM→ M exp p : T p M → M is a diffeomorphism. Then, we use this fact to present a formula, built on |ξ | | 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 .