Generalized complex structure: Difference between revisions

Revision as of 12:13, 5 May 2023

Revision as of 12:34, 5 May 2023

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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.

The fibers are endowed with a natural [[inner product]] with [[Metric signature (topology)|signature]] (''N'',&nbsp;''N''). If ''X'' and ''Y'' are vector fields and ''ξ'' and ''η'' are one-forms then the inner product of ''X+ξ'' and ''Y+η'' is defined as

:<math>\langle X+\xi,Y+\eta\rangle=\frac{1}{2}(\xi(Y)+\eta(X)).</math>