Closed conformal vector fields and Lagrangian submanifolds in complex space forms

We study a wide family of Lagrangian submanifolds in non-flat complex space forms that we will call pseudoumbilical be-cause 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 empha-size the case X = JH, where H is the mean curvature vector of the immersion, which are known as Lagrangian submani-folds with conformal Maslov form. In this family we offer different global characterizations of the Whitney spheres in the complex projective and hyperbolic spaces. Let Mn 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 manifoldM 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 must take into account the nice property of the second funda-mental form σ of these submanifolds which says that the trilinear form 〈σ(v, w), Jz〉 is totally symmetric. Sometimes this property of the second fundamental form gives obstructions to the existence of examples satisfying classical Rie-mannian properties. So, if one considers like classical property the umbilic-ity, automatically our umbilical Lagrangian submanifold is totally geodesic and only appear trivial examples. Then two natural questions arise: What is the Lagrangian version of umbilicity? Which are the corresponding exam-ples?. In [RU] it was proposed that the Lagrangian version of umbilicity was that the second fundamental form satisfies 〈σ(v, w), Jz 〉 = n n+ 2 ∂v,w,z〈v, w〉〈H,Jz〉,(0.1