Canonical connections attached to hypercomplex and biparacomplex structures

We put into light some generalized almost hypercomplex and almost biparacomplex structures and characterize their integrability with respect to a ∇-bracket on the generalized tangent bundle TM ⊕ T∗M of a smooth manifold M, defined by an affine connection ∇ on M. Also, we provide necessary and sufficient conditions for these structures to be ∇ˆ -parallel and ∇ˆ ∗ -parallel, where ∇ˆ is a generalized connection on M induced by ∇, and ∇ˆ ∗ is its generalized dual connection with respect to a bilinear form hˇ on TM ⊕ T∗M induced by a non-degenerate symmetric or skew-symmetric (0, 2)-tensor field h on M. As main results, we establish the existence of a canonical con- nection associated to a generalized hypercomplex and to a generalized biparacomplex structure, i.e., a torsion-free generalized connection that parallelizes these structures. We show that, in the hypercomplex case, the canonical connection is the generalized Obata connection and that on a quasi-statistical manifold (M, h, ∇), an integrable h-symmetric and ∇-parallel (1, 1)-tensor field gives rise to a generalized biparacom- plex structure whose canonical connection is precisely ∇ˆ ∗ . Finally we prove that the generalized connection that parallelizes certain families of generalized almost complex and almost product structures is preserved.