Opuscula Math. 41, no. 2 (2021), 205-226
https://doi.org/10.7494/OpMath.2021.41.2.205

Opuscula Mathematica

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

Włodzimierz M. Mikulski

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.

Keywords: natural operator, linear vector field, linear form, twisted Dorfman-Courant bracket, the Jacobi identity in Leibniz form.

Mathematics Subject Classification: 53A55, 53A45, 53A99.

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• Communicated by P.A. Cojuhari.
• Accepted: 2021-01-27.
• Published online: 2021-03-17.