Editing Almost complex manifold (section)

== Generalized almost complex structure ==
[[Nigel Hitchin]] introduced the notion of a [[generalized almost complex structure]] on the manifold ''M'', which was elaborated in the doctoral dissertations of his students [[Marco Gualtieri]] and [[Gil Cavalcanti]]. An ordinary almost complex structure is a choice of a half-dimensional [[Linear subspace|subspace]] of each fiber of the complexified [[tangent bundle]] ''TM''. A generalized almost complex structure is a choice of a half-dimensional [[Isotropic manifold|isotropic]] subspace of each fiber of the [[direct sum of vector bundles|direct sum]] of the complexified tangent and [[cotangent bundle]]s. In both cases one demands that the direct sum of the [[subbundle]] and its [[complex conjugate]] yield the original bundle.

An almost complex structure integrates to a complex structure if the half-dimensional subspace is closed under the [[Lie derivative|Lie bracket]]. A generalized almost complex structure integrates to a [[generalized complex structure]] if the subspace is closed under the [[Courant bracket]]. If furthermore this half-dimensional space is the annihilator of a nowhere vanishing [[pure spinor]] then ''M'' is a [[generalized Calabi–Yau manifold]].