Generalized para-Kähler manifolds

We define a generalized almost para-Hermitian structure to be a commuting pair (F,J) of a generalized almost para-complex structure and a generalized almost complex structure with an adequate non-degeneracy condition. If the two structures are integrable the pair is called a generalized para-Kähler structure. This class of structures contains both the classical para-Kähler structure and the classical Kähler structure. We show that a generalized almost para-Hermitian structure is equivalent to a triple (γ, ψ, F), where γ is a (pseudo) Riemannian metric, ψ is a 2-form and F is a complex (1, 1)-tensor field such that F²=Id, γ(FX, Y)+γ(X, FY)=0. We deduce integrability conditions similar to those of the generalized Kähler structures and give several examples of generalized para-Kähler manifolds. We discuss submanifolds that bear induced para-Kähler structures and, on the other hand, we define a reduction process of para-Kähler structures.