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

Opuscula Mathematica

article

Włodzimierz M. Mikulski¹ 

[1] Jagiellonian University, Department of Mathematics

关键词: natural operator;    linear vector field;    linear form;    twisted Dorfman-Courant bracket;    the Jacobi identity in Leibniz form.;   

DOI  :  10.7494/OpMath.2021.41.2.205

学科分类：环境科学（综合）

来源: AGH University of Science and Technology Press

【 摘 要 】

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.

【 授权许可】

CC BY-NC