Metric polynomial structures on generalized geometry

In this document, we study the interaction between different geometric structures defined on the generalized tangent bundle $\mathbb TM := TM\oplus T^*M\to M$. Specifically, we study various generalized polynomial $\alpha$-structures $\mathcal J$ with respect to different metrics $\mathcal G$ on $\mathbb TM$ when $\mathcal J^2 = \alpha \mathcal Id$, $\mathcal G(\mathcal J\cdot, \mathcal J\cdot) = \varepsilon\mathcal G(\cdot, \cdot)$, for $\alpha, \varepsilon\in \{+1, -1\}$. Besides, we study the commutation or anti-commutation of generalized polynomial structures, showing that a generalized Kähler structure can be understood as a pair of commuting generalized almost complex structures.