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A '''generalized almost complex structure''' is just an [[almost complex structure]] of the generalized tangent bundle which preserves the natural inner product:
:::<math>{\mathcal J}: \mathbf{T}\oplus\mathbf{T}^*\rightarrow \mathbf{T}\oplus\mathbf{T}^*</math>
such that :::<math>{\mathcal J}^2=-{\rm Id}, \mbox{ and }\langle {\mathcal J}(X+\xi),{\mathcal J}(Y+\eta)\rangle=\langle X+\xi,Y+\eta\rangle</math>▼
such that
▲
Like in the case of an ordinary [[almost complex structure]], a generalized almost complex structure is uniquely determined by its <math>\sqrt{-1}</math>-[[eigenbundle]], i.e. a subbundle <math>L</math> of the complexified generalized tangent bundle <math>(\mathbf{T}\oplus\mathbf{T}^*)\otimes\mathbb{C}</math>
given by
:::<math>L=\{X+\xi\in (\mathbf{T}\oplus\mathbf{T}^*)\otimes\mathbb{C}\ :\ {\mathcal J}(X+\xi)=\sqrt{-1}(X+\xi)\}</math>
Such subbundle ''L'' satisfies the following properties:
(i) the intersection with its [[complex conjugate]] is the zero section: <math>L\cap\overline{L}=0</math>;
(ii) ''L'' is '''maximally isotropic''', i.e. its complex [[rank]] equals ''N'' and <math>\langle\ell,\ell'\rangle=0</math> for all <math>\ell,\ell'\in L.</math>
Viceversa, any subbundle ''L'' satisfying (i), (ii) is the <math>\sqrt{-1}</math>-[[eigenbundle]] of a unique generalized complex structure, so that the properties (i), (ii) can be considered as an alternative definition of generalized almost complex structure.
===Courant bracket===
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