Generalized almost product structures and generalized CRF-structures

Abstract

We give several equivalent characterizations of orthogonal subbundles of the generalized tangent bundle defined, up to B-field transform, by almost product and local product structures. We also introduce a pure spinor formalism for generalized CRF-structure and investigate the resulting decomposition of the de Rham operator. As applications we give a characterization of generalized complex manifolds that are locally the product of generalized complex factors and discuss infinitesimal deformations of generalized CRF-structures.

Generalized symplectic Golden manifolds and Lie Groupoids

“…The above lemma shows that E (1,0) is not always isotropic. Next result gives necessary conditions for E (1,0) to be isotropic.…”

“…Proof. If E (1,0) and E (0,1) are eigenbundles corresponding to ϕ and 1 − ϕ, then E (1,0) ∩ E (0,1) = {0}. Suppose that E (1,0) is maximal isotropic.…”

“…After Hitchin's definition of generalized complex manifolds, this topic was studied in detail by Gualtieri [21] in his doctoral thesis, see also [20]. After this stage, new generalized manifolds were studied by many authors, see: [1], [3], [4], [5], [18], [32], [36], [41].…”

On hypersurfaces of generalized Kähler manifolds

“…The vanishing of the right hand side becomes the second condition which holds for any vector fields on a Hermitian manifold, and, in the end, replace Y by JY . (Identity (3.12) may be checked on any combination of arguments of the complex type (1, 0), (0, 1) while recalling that dΩ has only terms of type (2, 1), (1,2). )…”

“…The (2, 1)-generalized almost contact structures were also introduced under a different form in [13], where the author looks at E ± = (1/2)(Z + ± Z − ) rather than at Z ± . Notice that, if (F , Z ± ) is a (2, 1)-generalized almost contact structure, L = im F is a split structure in the sense of [1,2].…”