Calabi-Yau Versus Calabi-Yau Metric
Notice that a generalized Calabi metric structure, which was introduced by Gualtieri, is a stronger condition than a generalized Calabi-Yau structure, which was introduced by Hitchin. In particular a generalized Calabi-Yau metric structure implies the existence of two commuting generalized almost complex structures.
Read more about this topic: Generalized Complex Structure, Relation To G-structures