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===Classification===
There is a one-to-one correspondence between maximal isotropic [[subbundle]] of <math>\mathbf{T} \oplus \mathbf{T}^*</math> and pairs <math>('''\mathbf{E'''}, ''ε''\varepsilon)</math> where '''E''' is a subbundle of '''T''' and ''ε''<math>\varepsilon</math> is a 2-form. This correspondence extends straightforwardly to the complex case.
Given a pair <math>('''\mathbf{E'''},''ε'' \varepsilon)</math> one can construct a maximally isotropic subbundle ''<math>L''('''\mathbf{E'''},''ε'' \varepsilon)</math> of <math>\mathbf{T} \oplus \mathbf{T}^*</math> as follows. The elements of the subbundle are the [[formal sum]]s ''<math>X'' + ''ξ''\xi</math> where the [[vector field]] ''X'' is a section of '''E''' and the one-form ''ξ'' restricted to the [[dual space]] '''E'''<supmath>\mathbf{E}^*</supmath> is equal to the one-form ε<math>\varepsilon(''X'').</math>
To see that ''<math>L''('''\mathbf{E'''}, ''ε'' \varepsilon)</math> is isotropic, notice that if ''Y'' is a section of '''E''' and ''ξ''<math>\xi</math> restricted to <math>\mathbf{E}^*</math> is ''ε''<math>\varepsilon(''X'')</math> then ''ξ''<math>\xi(''Y'') = ''ε''\varepsilon(''X'', ''Y''),</math> as the part of ''ξ''<math>\xi</math> orthogonal to <math>\mathbf{E}^*</math> annihilates ''Y''. Thesefore if ''<math>X'' + ''ξ''\xi</math> and ''<math>Y'' + ''η''\eta</math> are sections of <math>\mathbf{T} \oplus \mathbf{T}^*</math> then
:<math>\langle X+\xi,Y+\eta\rangle=\frac{1}{2}(\xi(Y)+\eta(X))=\frac{1}{2}(\epsilonvarepsilon(Y,X)+\epsilonvarepsilon(X,Y))=0</math>
and so ''<math>L''('''\mathbf{E'''}, ''ε''\varepsilon)</math> is isotropic. Furthermore, ''<math>L''('''\mathbf{E'''}, ''ε''\varepsilon)</math> is maximal because there are <math>\dim('''\mathbf{E'''})</math> (complex) dimensions of choices for '''<math>\mathbf{E'''},</math> and ''ε''<math>\varepsilon</math> is unrestricted on the [[complement (complexity)|complement]] of <math>\mathbf{E}^*,</math> which is of (complex) dimension ''<math>n'' − -\dim('''\mathbf{E'''}).</math> Thus the total (complex) dimension in ''n''. Gualtieri has proven that all maximal isotropic subbundles are of the form ''<math>L''('''\mathbf{E'''}, ''ε''\varepsilon)</math> for some '''<math>\mathbf{E'''}</math> and ''ε''<math>\varepsilon.</math>
===Type===
The '''type''' of a maximal isotropic subbundle ''<math>L''('''\mathbf{E'''}, ''ε''\varepsilon)</math> is the real dimension of the subbundle that annihilates '''E'''. Equivalently it is 2''N'' minus the real dimension of the [[projection (mathematics)|projection]] of ''<math>L''('''\mathbf{E'''}, ''ε''\varepsilon)</math> onto the tangent bundle '''T'''. In other words, the type of a maximal isotropic subbundle is the codimension of its projection onto the tangent bundle. In the complex case one uses the complex dimension and the type is sometimes referred to as the '''complex type'''. While the type of a subbundle can in principle be any integer between 0 and 2''N'', generalized almost complex structures cannot have a type greater than ''N'' because the sum of the subbundle and its complex conjugate must be all of <math>(\mathbf{T} \oplus \mathbf{T}^*) \otimes \C.</math>
The type of a maximal isotropic subbundle is [[Invariant (mathematics)|invariant]] under [[diffeomorphisms]] and also under shifts of the [[Kalb-Ramond field|B-field]], which are [[isometry|isometries]] of <math>\mathbf{T} \oplus \mathbf{T}^*</math> of the form
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