Metric polynomial structures on generalized geometry

In this document, we study the interaction between different geometric structures that can be defined as morphisms of sections of the generalized tangent bundle $\mathbb TM:= TM\oplus T^*M\to M$. In particular, we show the behaviour of various generalized polynomial structures with respect to different metrics on $\mathbb TM$, in the frame of $(\alpha, \varepsilon)$-metric structures; and the commutation or anti-commutation of generalized polynomial structures, forming triple structures.