 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 formula_48 Generalized Kähler structures are reductions of the structure group to formula_76 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 formula_77

Notice that a generalized Calabi metric structure, which was introduced by Marco Gualtieri, is a stronger condition than a generalized Calabi–Yau structure, which was introduced by Nigel Hitchin.