A rigidity result for Kählerian manifolds endowed with closed conformal vector fields

Antonio Caminha

doi:10.33044/revuma.v60n2a13

Abstract

We show that if a connected compact K¨ahlerian surface M with nonpositive Gaussian curvature is endowed with a closed conformal vector field ξ whose singular points are isolated, then M is isometric to a flat torus and ξ is parallel. We also consider the case of a connected complete K¨ahlerian manifod M of complex dimension n > 1 and endowed with a nontrivial closed conformal vector field ξ. In this case, it is well known that the singularities of ξ are automatically isolated and the nontrivial leaves of the distribution generated by ξ and Jξ are totally geodesic in M. Assuming that one such leaf is compact, has torsion normal holonomy group and that the holomorphic sectional curvature of M along it is nonpositive, we show that ξ is parallel and M is foliated by a family of totally geodesic isometric tori and also by a family of totally geodesic isometric complete K¨ahlerian manifolds of complex dimension n − 1. In particular, the universal covering of M is isometric to a Riemannian product having R2 as a factor. We also comment on a generic class of compact complex symmetric spaces not possessing nontrivial closed conformal vector fields, thus showing that we cannot get rid of the hypothesis of nonpositivity of the holomorphic sectional curvature in the direction of ξ.

Highlights

A conformal vector field ξ on a semi-Riemannian manifold M is closed if its metrically dual 1-form is closed
In both of these classes of examples, the holomorphic sectional curvature of M in the direction of ξ vanishes identically and, if J stands for the quasi-complex structure of M, the leaves of the distribution generated by ξ and Jξ are totally geodesic in M
The purpose of this paper is to show that, under a reasonable set of conditions on the closed conformal vector field ξ, the second class of examples presented in the third paragraph is essentially the only one

Summary

Introduction

A conformal vector field ξ on a semi-Riemannian manifold M is closed if its metrically dual 1-form is closed. The geometry of Riemannian submanifolds of Lorentzian and Riemannian manifolds in which either the submanifold or the ambient space is endowed with a closed conformal vector field has been the object of intense research in recent years (see, for instance, [1, 3, 6, 7] and the references therein). The presence of such a (nontrivial) vector field imposes strong restrictions on the structure of the ambient manifold M itself. Kahlerian metrics; Closed conformal vector fields; Geometric foliations; Symmetric spaces

CAMINHA

Deforming Kahlerian metrics

Two rigidity results for Kahlerian manifolds