Closed conformal vector fields and Lagrangian submanifolds in complex space forms

Abstract

We study a wide family of Lagrangian submanifolds in non flat complex space forms that we will call pseudoumbilical because of their geometric properties. They are determined by admitting a closed and conformal vector field X such that X is a principal direction of the shape operator AJX , being J the complex structure of the ambient manifold. We emphasize the case X = JH, where H is the mean curvature vector of the immersion, which are known as Lagrangian submanifolds with conformal Maslov form. In this family we offer different global characterizations of the Whitney spheres in the complex projective and hyperbolic spaces. Let M be a Kaehler manifold of complex dimension n. The Kaehler form Ω on M is given by Ω(v, w) = 〈v, Jw〉, being 〈, 〉 the metric and J the complex structure on M . An immersion φ : M −→ M of an n-dimensional manifold M is called Lagrangian if φ∗Ω ≡ 0. This property involves only the symplectic structure of M . In this family of Lagrangian submanifolds, one can study properties of the submanifold involving the Riemannian structure of M . One ∗Research partially supported by a DGICYT grant No. PB97-0785.