Characterization of Warped Product Lagrangian Submanifolds in Cⁿ

Abstract

The procedure of constructing a Lagrangian immersion in the complex projective space, starting with two other Lagrangian immersions into complex projective spaces of lesser dimension is known as a Calabi product, motivated by the similar construction in the affine differential geometry. In particular, one may consider a point instead of the one of the immersions, and in both cases the submanifold has a warped product structure of the interval and one or two Lagrangian immersions. Such Lagrangian submanifold then admits a splitting of the tangent bundle into orthogonal subbundles defined in terms of the corresponding second fundamental form, in case of a point and an immersion decomposition consists of two components and in case of a proper Calabi product, decomposition has three components. The generalization of this notion was investigated for Lagrangian immersions in complex space form Mn(4 c) , where c≠ 0. Here we study the case c= 0. We investigate the properties of the Lagrangian immersions into Cn with tangent bundle admitting the decomposition in question and further, we give explicit expressions for such immersions.