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 M$M$ endowed with a torsion free linear connection, ∇$\nabla$. We introduce the concept of ∇$\nabla$-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 M$M$ induced by complex structures on the generalized tangent bundle of M$M$.