Relation To G-structures
Some of the almost structures in generalized complex geometry may be rephrased in the language of G-structures. The word "almost" is removed if the structure is integrable.
The bundle (TT*) C with the above inner product is a O(2n, 2n) structure. A generalized almost complex structure is a reduction of this structure to a U(n, n) structure. Therefore the space of generalized complex structures is the coset
A generalized almost Kähler structure is a pair of commuting generalized complex structures such that minus the product of the corresponding tensors is a positive definite metric on (T T*)C. Generalized Kähler structures are reductions of the structure group to U(n)U(n). Generalized Kähler manifolds, and their twisted counterparts, are equivalent to the bihermitian manifolds discovered by Sylvester James Gates, Chris Hull and Martin Roček in the context of 2-dimensional supersymmetric quantum field theories in 1984.
Finally, a generalized almost Calabi-Yau metric structure is a further reduction of the structure group to SU(n)SU(n).
Read more about this topic: Generalized Complex Structure
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