Almost complex structures on cotangent bundles and generalized geometry

We study a class of complex structures on the generalized tangent bundle of a smooth manifold MM endowed with a torsion free linear connection, ∇∇. We introduce the concept of ∇∇-integrability and we study integrability conditions. In the case of the generalized complex structures introduced by Hitchin (2003) in [2], we compare the two concepts of integrability. Moreover, as an application, we describe almost complex structures on the cotangent bundle of MM induced by complex structures on the generalized tangent bundle of MM.