Generalized para-Kähler manifolds

We define a generalized almost para-Hermitian structure to be a commuting pair (F,J)(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)(γ,ψ,F), where γ is a (pseudo) Riemannian metric, ψ is a 2-form and F   is a complex (1,1)(1,1)-tensor field such that F2=IdF2=Id, γ(FX,Y)+γ(X,FY)=0γ(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.