Generalized para-Kähler manifolds

We define a generalized almost para-Hermitian structure to be a commuting pair $(\mathcal{F},\mathcal{J})$ 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 $F^2=Id,γ(FX,Y)+γ(X,FY)=0$. 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.