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 subspace of each fiber of the complexified tangent bundle TM. A generalized almost complex structure is a choice of a half-dimensional isotropic subspace of each fiber of the direct sum of the complexified tangent and cotangent bundles. 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 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.
Read more about this topic: Almost Complex Manifold
Famous quotes containing the words generalized, complex and/or structure:
“One is conscious of no brave and noble earnestness in it, of no generalized passion for intellectual and spiritual adventure, of no organized determination to think things out. What is there is a highly self-conscious and insipid correctness, a bloodless respectability submergence of matter in manner—in brief, what is there is the feeble, uninspiring quality of German painting and English music.”
—H.L. (Henry Lewis)
“All of life and human relations have become so incomprehensibly complex that, when you think about it, it becomes terrifying and your heart stands still.”
—Anton Pavlovich Chekhov (1860–1904)
“A special feature of the structure of our book is the monstrous but perfectly organic part that eavesdropping plays in it.”
—Vladimir Nabokov (1899–1977)