On a class of pseudo calibrated generalized complex structures related to Norden, para-Norden and statistical manifolds

We consider pseudo calibrated generalized complex structures, J, defined by a pseudo Riemannian metric g and a g-symmetric operator H such that H²=μI, μ∈ℝ, on a smooth manifold M. These structures include the case of complex Norden manifolds for μ=-1, studied inprevious work, the case of almost tangent structures for μ=0, ImH=kerH, and the case of para Norden manifolds for μ=1. The special case H=O is described in previous work. We study first the integrability conditions of J, with respect to a linear connection ∇, and we describe examples of geometric structures that naturally give rise to integrable pseudo calibrated generalized complex structures. Then we prove that for μ≠-1 integrability implies that the ±i-eigenbundles of J, E_{J}^{1,0}, E_{J}^{0,1}, are complex Lie algebroids. Moreover we define the concept of generalized ∂_{J}-operator of (M,H,g,∇) and we study holomorphic sections. This paper is a generalization of our previous papers N4, N5 and allows us to unify complex Norden and para Norden manifolds trough almost tangent structures and statistical manifolds. The theory reveals that the case of complex Norden manifolds is special.