Para-Kähler Immersions in Para-Kähler Space Forms

Abstract A classical problem addressed, among others, by Calabi, is that to characterize non-isometric Kähler manifolds immersed in a finite-dimensional Kähler space form. In this paper we address the same problem in the para-Kähler context and provide necessary and sufficient conditions for the existence of para-Kähler immersions in para-Kähler space forms. As a consequence, we prove that, in general, a local para-Kähler immersion cannot be globally extended, even if it is defined on a simply connected para-Kähler manifold. Finally, we classify para-Kähler immersions between para-Kähler space forms.