Generalized geometry of pseudo-Riemannian manifolds and the generalized (partial derivative)over-bar-operator

摘要

Let (M, g, del) be a pseudo-Riemannian manifold with a torsion free linear connection and let J(g) be the generalized complex structure on M defined by g; see [13], [14]. We prove that in the case J(g) is del integrable the +/- i-eigenbundles of J(g), E-Jg(1,0) and E-Jg(1,0) are complex Lie algebroids. Moreover E-Jg(1,0) and (E-Jg(1,0))* are canonically isomorphic, thus we define the concept of generalized (partial derivative) over bar -operator of (M, g, del) and we describe a class of generalized holomorphic sections of T(M) circle plus T* (M). Also we relate the Lie bialgebroid property of (E-Jg(1,0), (E-Jg(1,0))*) to conditions on the metric g in the case of affine Hessian manifolds.