On complete Kählerian manifolds endowed with closed conformal vector fields

Abstract

Let {overline{M}}^{2n}, n>1, be a complete, noncompact Kählerian manifold, endowed with a nontrivial closed conformal vector field xi having at least one singular point. Under a reasonable set of conditions, we show that xi has just one singular point p and that {overline{M}}{setminus }{p} is isometric to a one dimensional cone over a simply connected Sasakian manifold N diffeomorphic to {mathbb {S}}^{2n-1}.As a straightforward consequence, we conclude that if the addition of a single point to the Kählerian cone of a (2n-1)-dimensional Sasakian manifold N has the structure of a complete, noncompact, 2n-dimensional Kählerian manifold whose metric extends that of the cone, and such that the canonical vector field of the cone extends to it having a singularity at the extra point, then N is isometric to mathbb S^{2n-1}, endowed with an appropriate Sasakian structure.