Generalized geometry of Norden manifolds

Abstract

Let ( M , J , g , D ) be a Norden manifold with the natural canonical connection D and let J ̂ be the generalized complex structure on M defined by g and J . We prove that J ̂ is D -integrable and we find conditions on the curvature of D under which the ± i -eigenbundles of J ̂ , E J ̂ 1 , 0 , E J ̂ 0 , 1 , are complex Lie algebroids. Moreover we proove that E J ̂ 1 , 0 and ( E J ̂ 1 , 0 ) ∗ are canonically isomorphic and this allow us to define the concept of generalized ∂ ¯ J ̂ -operator of ( M , J , g , D ) . Also we describe some generalized holomorphic sections. The class of Kähler–Norden manifolds plays an important role in this paper because for these manifolds E J ̂ 1 , 0 and E J ̂ 0 , 1 are complex Lie algebroids.