The Natural Operators Similar to the Twisted Courant Bracket One

Given natural numbers m≥3\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$m\ge 3$$\end{document} and p≥3\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$p\ge 3$$\end{document}, all Mfm\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal {M} f_m$$\end{document}-natural operators AH\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$A_{H}$$\end{document} sending p-forms H∈Ωp(M)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$H\in \Omega ^p(M)$$\end{document} on m-manifolds M into bilinear operators AH:(X(M)⊕Ω1(M))×(X(M)⊕Ω1(M))→X(M)⊕Ω1(M)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$A_H:(\mathcal {X}(M)\,\oplus \,\Omega ^1(M))\times (\mathcal {X}(M)\,\oplus \,\Omega ^1(M))\rightarrow \mathcal {X}(M)\,\oplus \,\Omega ^1(M)$$\end{document} transforming pairs of couples of vector fields and 1-forms on M into couples of vector fields and 1-forms on M are founded. If m≥3\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$m\ge 3$$\end{document} and p≥3\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$p\ge 3$$\end{document}, then that any (similar as above) Mfm\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal {M} f_m$$\end{document}-natural operator A which is defined only for closed p-forms H can be extended uniquely to the one A which is defined for all p-forms H is observed. If p=3\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$p=3$$\end{document} and m≥3\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$m\ge 3$$\end{document}, all Mfm\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal {M} f_m$$\end{document}-natural operators A (as above) such that AH\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$A_H$$\end{document} satisfies the Leibniz rule for all closed 3-forms H on m-manifolds M are extracted. The twisted Courant bracket [-,-]H\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$[-,-]_H$$\end{document} for all closed 3-forms H on m-manifolds M gives the most important example of such Mfm\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal {M} f_m$$\end{document}-natural operator A.


Introduction
The "doubled" tangent bundle T ⊕ T * over m-dimensional manifolds (mmanifolds) is full of interest because it has the natural inner product, and the Courant bracket, see [1]. Besides, generalized complex structures are defined on T ⊕ T * , generalizing both (usual) complex and symplectic structures, see e.g. [3,4].
The Courant bracket [−, −] C from Example 2.2 does not satisfy the Leibniz rule.
Theorem 2.7 [2]. If m ≥ 2, any Mf m -natural bilinear operator A in the sense of Definition 2.1 satisfying the Leibniz rule is one of the following ones: where a is an arbitrary real number, and where ρ 1 = X 1 ⊕ ω 1 and ρ 2 = X 2 ⊕ ω 2 .
Of course, the anchor a : where v i are the coordinates of v and ω j are the coordinates of ω, and where α i and β j are the real numbers determined by a 0 . Then using the invariance of a 0 with respect to the maps (τ 1 x 1 , ..., τ m x m ) for τ 1 > 0, ..., τ m > 0 we deduce that α 2 = · · · = α m = 0 and β 1 = · · · = β m = 0. Then the vector space of all a in question is at most 1-dimensional. Thus the dimension argument completes the proof.
On the other hand one can directly show that The

The Natural Operators Similar to the Twisted Courant Bracket
The regularity of A means that it transforms smoothly parametrized families (H t ,  The main result of this section is the following   A) reals a, b 1 , ..., c, where 3

for (uniquely determined by
By the non-linear Peetre theorem, see [5], A is of finite order. It means that there is a finite number r such that from (j r So, we may assume that H, X 1 , X 2 , ω 1 , ω 2 are polynomials of degree not more than r. Using the invariance of A with respect to the homotheties and the bilinearity of A H (for given H) we obtain homogeneity condition Then, by the homogeneous function theorem, since A is of finite order and regular and A 0 = 0 and p ≥ 2, we have A 1 Using the same arguments we get homogeneity condition Then, if p = 2, by the homogeneous function theorem and the bilinearity of A H and the assumptions A 0 = 0 and A H = A H+dH 1 , the value H |0 ) and μ, only. By m ≥ 3 and the regularity of A, we may assume that X 1 |0 , X 2 |0 and μ are linearly independent. Then by the invariance we may assume Then using the invariance of A with respect to τ id for τ i > 0 we deduce that only v := A 2 Therefore the vector space of all A in question with A 0 = 0 and A H = A H+dH 1 is at most one-dimensional. The part (1) of the theorem is complete. If p = 3, then (by almost the same arguments as for Then using the invariance with respect to (τ 1 x 1 , ...τ m x m ) for τ i > 0 we deduce that only the value A 2 dx 1 ∧dx 2 ∧dx 3 (∂ 1 ⊕ 0, ∂ 2 ⊕ 0), ∂ 3|0 ∈ R may be not-zero. Therefore the vector space of all A in question with A 0 = 0 is one-dimensional (generated by the natural operator 0 ⊕ i X 1 i X 2 H).
If p ≥ 4, then (similarly as for p = 2) < A 2 H (X 1 ⊕ω 1 , X 2 ⊕ω 2 ) |0 , μ >= 0. Theorem 3.4 is complete.  uniquely determined by A) real numbers a, b, c, e. Roughly speaking, Corollary 3.5 says that any Mf m -natural operator A in the sense of Definition 3.1 such that A H is skew-symmetric for any H ∈ Ω 3 (M ) and any m-manifold M coincides with the "skew-symmetrization" of the twisted Courant bracket Mf m -natural operator up to four real constants a, b, c, e.   determined reals a, b 1 , ..., c such that for all H ∈ Ω 3 (M ) and m-manifolds M

The Natural Operators Similar to the Twisted Courant Bracket and Defined for Closed p-Forms Only
In the previous section, we considered Mf m -natural operators A which are defined for all p-forms H. In this section, we observe what happens if A are defined for closed p-forms H, only. We start with the following