Closed conformal vector fields and Lagrangian submanifolds in complex space forms

Ildefonso Castro, Cristina R. Montealegre and Francisco Urbano

Vol. 199 (2001), No. 2, 269–302

DOI: 10.2140/pjm.2001.199.269

Abstract

We study a wide family of Lagrangian submanifolds in nonflat 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 A_JX, 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.

Milestones

Received: 13 September 1999

Revised: 31 March 2000

Published: 1 June 2001