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===The generalized tangent bundle===
Consider an [[Manifold|''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#Sections|section]] of '''T''' is a [[vector field]] on ''M''. The [[cotangent bundle]] of ''M'', denoted '''T'''<sup>*</sup>, 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#Exterior power|exterior powers]] of the cotangent bundle. Generalized geometry unites these two fields by treating sections of the '''generalized tangent bundle''', which is the [[direct sum of vector bundles|direct sum]] <math>\mathbf{T} \oplus \mathbf{T}^*</math> of the tangent and cotangent bundles, which are formal sums of a vector field and a one-form.
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