The natural operators similar to the twisted Courant bracket on couples of vector fields and $p$-forms

Abstract

 Given natural numbers $m$ and $p$ with $m\geq p+2\geq 3$, all $\mathcal{M} f_m$-natural operators $A$ sending closed $(p+2)$-forms $H$ on $m$-manifolds $M$ into $\mathbf{R}$-bilinear operators $A_H$ transforming pairs of couples of vector fields and $p$-forms on $M$ into couples of vector fields and $p$-forms on $M$ are found. If $m\geq p+2\geq 3$ ,  all  $\mathcal{M} f_m$-natural operators $A$ (as above) such that $A_H$ satisfies the Jacobi identity in Leibniz form are extracted, and that the twisted Courant bracket  $[-,-]_H$ is the unique $\mathcal{M} f_m$-natural operator $A_H$ (as above) satisfying the Jacobi identity in Leibniz form and some normalization condition is deduced