Generalized complex structure: Difference between revisions

Revision as of 19:53, 20 May 2010

Revision as of 16:20, 7 June 2010

Line 5:

==Definition==

===The tangent plus cotangent bundle===

Consider an ''N''-manifold ''M''. The [[tangent bundle]] of ''M'', which will be denoted '''T''', is the [[vector bundle]] over ''M'' whose fibers consist of all [[tangent vector]]s to ''M''. A [[fiber bundle#SectionSections|section]] of '''T''' is a [[vector field]] on ''M''. The [[cotangent bundle]] of ''M'', denoted '''T'''^*, is the vector bundle over ''M'' whose sections are [[differential form|one-forms]] on ''M''.

In [[complex geometry]] one considers structures on the tangent bundles of manifolds. In [[symplectic geometry]] one is instead interested in [[Exterior algebra#The exterior power|exterior powers]] of the cotangent bundle. Generalized complex geometry unites these two fields by treating sections of the [[direct sum of vector bundles|direct sum]] ('''T'''&nbsp;<math>\oplus</math>