Generalized para-K\"ahler 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-K\"ahler structure. This class of structures contains both the classical para-K\"ahler structure and the classical K\"ahler structure. We show that a generalized almost para-Hermitian structure is equivalent to a triple $(\gamma,\psi,F)$, where $\gamma$ is a (pseudo) Riemannian metric, $\psi$ is a $2$-form and $F$ is a complex $(1,1)$-tensor field such that $F^2=Id,\gamma(FX,Y)+\gamma(X,FY)=0$. We deduce integrability conditions similar to those of the generalized K\"ahler structures and give several examples of generalized para-K\"ahler manifolds. We discuss submanifolds that bear induced para-K\"ahler structures and, on the other hand, we define a reduction process of para-K\"ahler structures.