Generalized geometry of pseudo Riemannian manifolds and generalized debar operator

Let (M;g) be a pseudo-Riemannian manifold with a torsion free linear connection and let Jg be the generalized complex structure on M defined by g. We prove that in the case Jg is integrable the i and -i eigenbundles of Jg are complex Lie algebroids. Moreover we define the concept of generalized de bar operator and we describe a class of generalized holomorphic sections. Also we relate Lie bialgebroid property to conditions on the metric g in the case of Hessian manifolds.