On complete Kählerian manifolds endowed with closed conformal vector fields

Abstract

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