On the gauge-natural operators similar to the twisted Dorfman-Courant bracket

Abstract

All $\mathcal{VB}_{m,n}$-gauge-natural operators $C$ sending linear $3$-forms $H \in \Gamma^{l}_E(\bigwedge^3T^*E)$ on a smooth ($\mathcal{C}^\infty$) vector bundle $E$ into $\mathbf{R}$-bilinear operators $C_H:\Gamma^l_E(TE \oplus T^*E)\times \Gamma^l_E(TE \oplus T^*E)\to \Gamma^l_E(TE \oplus T^*E)$ transforming pairs of linear sections of $TE \oplus T^*E \to E$ into linear sections of $TE \oplus T^*E \to E$ are completely described. The complete descriptions is given of all generalized twisted Dorfman-Courant brackets $C$ (i.e. $C$ as above such that $C_0$ is the Dorfman-Courant bracket) satisfying the Jacobi identity for closed linear $3$-forms $H$. An interesting natural characterization of the (usual) twisted Dorfman-Courant bracket is presented.