Calabi Product Lagrangian Immersions in Complex Projective Space and Complex Hyperbolic Space

Abstract

Starting from two Lagrangian immersions and a Legendre curve γ̃(t) in 3 (1) (or in ℍ31(-1)), it is possible to construct a new Lagrangian immersion in ℂℙ n(4) (or in ℂℍn(-4)), which is called a warped product Lagrangian immersion. When, where r1, r2, and a are positive constants with r21 + r22 = 1 (or -r21 + r22 = -1, we call the new Lagrangian immersion a Calabi product Lagrangian immersion. In this paper, we study the inverse problem: how to determine from the properties of the second fundamental form whether a given Lagrangian immersion of ℂℙn(4) or ℂℍn(-4) is a Calabi product Lagrangian immersion. When the Calabi product is minimal, or is Hamiltonian minimal, or has parallel second fundamental form, we give some further characterizations. © 2011 Springer Basel AG.