Variedades com campos conformes fechados

Abstract

In this work, we study complete Riemannian manifolds (M,g) endowed with a nontrivial closed conformal fields ξ, with at least one singular point p and whose conformal factor ψ satisfies an additional hypothesis. More precisely, we assume that the integral of ψ along each nontrivial geodesic departing from p is positive. We prove a formula for the volume of the geodesic balls centered in p, which only depends on |ξ|, and characterize this class of manifolds. In the context of Riemannian surfaces, we deduce a criterion, dependening only on |ξ|, for the classification up to conformal equivalences. Finally, in the context of Kählerian manifolds, we prove that the only example, up to isometries, is the complex Euclidean space.