The Natural Operators Similar to the Twisted Courant Bracket One

Abstract

Given natural numbers $m\ge 3$ and $p\ge 3$, all $\mathcal {M} f_m$-natural operators $A_{H}$ sending p-forms $H\in \Omega ^p(M)$ on m-manifolds M into bilinear operators $A_H:(\mathcal {X}(M)\,\oplus \,\Omega ^1(M))\times (\mathcal {X}(M)\,\oplus \,\Omega ^1(M))\rightarrow \mathcal {X}(M)\,\oplus \,\Omega ^1(M)$ 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 $m\ge 3$ and $p\ge 3$, then that any (similar as above) $\mathcal {M} f_m$-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 $p=3$ and $m\ge 3$, all $\mathcal {M} f_m$-natural operators A (as above) such that $A_H$ satisfies the Leibniz rule for all closed 3-forms H on m-manifolds M are extracted. The twisted Courant bracket $[-,-]_H$ for all closed 3-forms H on m-manifolds M gives the most important example of such $\mathcal {M} f_m$-natural operator A.

1 Introduction

The “doubled” tangent bundle $T\,\oplus \, T^*$ over m-dimensional manifolds (m-manifolds) is full of interest because it has the natural inner product, and the Courant bracket, see [1]. Besides, generalized complex structures are defined on $T\,\oplus \, T^*$, generalizing both (usual) complex and symplectic structures, see e.g. [3, 4].

In Sect. 2, the description from [2] of all $\mathcal {M} f_m$-natural bilinear operators

$$\begin{aligned}A:(\mathcal {X}(M)\,\oplus \,\Omega ^1(M))\times (\mathcal {X}(M)\,\oplus \,\Omega ^1(M))\rightarrow \mathcal {X}(M)\,\oplus \,\Omega ^1(M),\end{aligned}$$

transforming pairs of couples of vector fields and 1-forms on m-manifolds M into couples of vector fields and 1-forms on M will be shortly cited. The most important example of such $\mathcal {M} f_m$-natural bilinear operator A is given by the Courant bracket $[-,-]^C$, see Example 2.2. This Courant bracket was used in [1] to define the concept of Dirac structures being hybrid of both symplectic and Poisson structures.

In Sect. 2 we also deduce that the “trivial” Lie algebroid $(TM\,\oplus \, T^*M,0,0)$ is the only $\mathcal {M} f_m$-natural Lie algebroid $(EM, [[-,-]],a)$ with $EM:=TM\,\oplus \, T^*M$.

In Sect. 3, using essentially the results from [2], if $m\ge 3$ and $p\ge 3$, we find all $\mathcal {M} f_m$-natural operators A sending p-forms $H\in \Omega ^p(M)$ on m-manifolds M into bilinear maps

$$\begin{aligned}A_H:({\mathcal {X}}(M)\,\oplus \,\Omega ^1(M))\times ({\mathcal {X}}(M)\,\oplus \,\Omega ^1(M))\rightarrow {\mathcal {X}}(M)\,\oplus \,\Omega ^1(M).\end{aligned}$$

The most important example of such A is given by the H-twisted Courant bracket $[-,-]_H$ for all 3-forms H on m-manifolds M, see Example 3.2. Properties of $[-,-]_H$ (as the Leibniz rule for closed 3-forms H) were used in [7, 8] to define the concept of exact Courant algebroid.

In Sect. 4, we observe that if $m\ge 3$ and $p\ge 3$, then any (similar as above) $\mathcal {M} f_m$-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.

In Sect. 5, if $p=3$ we extract all $\mathcal {M} f_m$-natural operators A as above satisfying the Leibniz rule

$$\begin{aligned}A_H(\rho _1,A_H(\rho _2,\rho _3))=A_H(A_H(\rho _1,\rho _2),\rho _3)+A_H(\rho _2,A_H(\rho _1,\rho _3)),\end{aligned}$$

for any closed $H\in \Omega ^3(M)$, $\rho _1,\rho _2,\rho _3\in {\mathcal {X}}(M)\,\oplus \,\Omega ^1(M)$ and $M\in obj(\mathcal {M} f_m)$.

From now on, $(x^i)$ ($i=1,...,m$) denote the usual coordinates on $\mathbf {R}^m$ and $\partial _i={\partial \over \partial x^i}$ are the canonical vector fields on $\mathbf {R}^m$.

All manifolds considered in this paper are assumed to be finite dimensional second countable Hausdorff without boundary and smooth (of class $\mathcal {C}^\infty$). Maps between manifolds are assumed to be smooth (of class $\mathcal {C}^\infty$)

2 The Natural Bilinear Operators Similar to the Courant Bracket

The general concept of natural operators can be found in the fundamental monograph [5]. In the paper, we need two particular cases of natural operators presented in Definitions 2.1 (below) and 3.1 (in the next section).

Let $\mathcal {M} f_m$ be the category of m-dimensional $\mathcal {C}^\infty$ manifolds as objects and their immersions of class $\mathcal {C}^\infty$ as morphisms ($\mathcal {M} f_m$-maps).

Definition 2.1

A natural (called also $\mathcal {M} f_m$-natural) operator A sending pairs of couples of vector fields and 1-forms on m-manifolds M into couples of vector fields and 1-forms on M is a $\mathcal {M} f_m$-invariant family of operators (functions)

$$\begin{aligned} A:(\mathcal {X}(M)\,\oplus \,\Omega ^1(M))\times (\mathcal {X}(M)\,\oplus \,\Omega ^1(M))\rightarrow \mathcal {X}(M)\,\oplus \,\Omega ^1(M),\end{aligned}$$

for all m-manifolds M, where $\mathcal {X}(M)\,\oplus \,\Omega ^1(M)$ is the vector space of couples $(X,\omega )$ of vector fields X on M and 1-forms $\omega$ on M. Such $\mathcal {M} f_m$-natural operator A is called bilinear if A is bilinear (i.e., $A(\rho ^1,-)$ and $A(-,\rho ^2)$ are linear (over the field $\mathbf {R}$ of real numbers) functions $\mathcal {X}(M)\,\oplus \,\Omega ^1(M)\rightarrow \mathcal {X}(M)\,\oplus \,\Omega ^1(M)$ for any fixed $\rho ^1,\rho ^2\in \mathcal {X}(M)\,\oplus \,\Omega ^1(M)$) for any m-manifold M. Such $\mathcal {M} f_m$-natural operator A is called skew-symmetric if A is skew-symmetric for any m-manifold M.

The $\mathcal {M} f_m$-invariance of A means that if $(X^1\,\oplus \,\omega ^1, X^2\,\oplus \,\omega ^2)$ and $(\overline{X}^1\,\oplus \,\overline{\omega }^1,\overline{X}^2\,\oplus \,\overline{\omega }^2)$ are $\varphi$-related by an $\mathcal {M} f_m$-map $\varphi :M\rightarrow \overline{M}$ (i.e., $\overline{X}^i\circ \varphi =T\varphi \circ X^i$ and $\overline{\omega }^i\circ \varphi =T^*\varphi \circ \omega ^i$ for $i=1,2$), then so are $A(X^1\,\oplus \,\omega ^1,X^2\,\oplus \,\omega ^2)$ and $A(\overline{X}^1\,\oplus \,\overline{\omega }^1,\overline{X}^2\,\oplus \,\overline{\omega }^2)$.

The most important example of such $\mathcal {M} f_m$-natural bilinear operator A is given by the (skew-symmetric) Courant bracket $[-,-]^C$ for any m-manifold M.

Example 2.2

On the vector bundle $TM\,\oplus \, T^*M$ there exist canonical symmetric and skew-symmetric pairings

$$\begin{aligned} \left<X^1\,\oplus \,\omega ^1,X^2\,\oplus \,\omega ^2\right>_\pm ={1\over 2}(i_{X^2}\omega ^1\pm i_{X^1}\omega ^2) \end{aligned}$$

for any $X^1\,\oplus \,\omega ^1, X^2\,\oplus \,\omega ^2\in \mathcal {X}(M)\,\oplus \,\Omega ^1(M)$, where i is the interior derivative. Further, the (skew-symmetric) Courant bracket is given by

$$\begin{aligned}&{[}X^1\,\oplus \,\omega ^1,X^2\,\oplus \,\omega ^2]^C\\&\quad =[X^1,X^2]\,\oplus \,\left( \mathcal {L}_{X^1}\omega ^2-\mathcal {L}_{X^2}\omega ^1+\mathrm {d}\left<X^1\,\oplus \,\omega ^1,X^2\,\oplus \, \omega ^2\right>_-\right) \end{aligned}$$

for any $X^1\,\oplus \,\omega ^1, X^2\,\oplus \,\omega ^2\in \mathcal {X}(M)\,\oplus \,\Omega ^1(M)$, where $[-,-]$ is the usual bracket on vector fields, $\mathcal {L}$ is the Lie derivative and d is the exterior derivative.

Theorem 2.3

[2]. If $m\ge 2$, any $\mathcal {M} f_m$-natural bilinear operator A in the sense of Definition 2.1 is of the form

$$\begin{aligned}&A (\rho ^1,\rho ^2) \\ {}&\quad = a[X^1,X^2]\,\oplus \, \left( b_1\mathcal {L}_{X^2}\omega ^1+b_2\mathcal {L}_{X^1}\omega ^2 +b_3\mathrm {d}\left<\rho ^1,\rho ^2\right>_+ + b_4\mathrm {d}\left<\rho ^1,\rho ^2\right>_-\right) \end{aligned}$$

for (uniquely determined by A) real numbers $a, b_1,b_2,b_3,b_4$, where $\rho ^i=X^i\,\oplus \,\omega ^i\in \mathcal {X}(M)\,\oplus \,\Omega ^1(M)$ for $i=1,2$ are arbitrary, and where $\left<-,-\right>_+$ and $\left<-,-\right>_-$ are as in Example 2.2.

Corollary 2.4

[2]. If $m\ge 2$, any $\mathcal {M} f_m$-natural skew-symmetric bilinear operator A in the sense of Definition 2.1 is of the form

$$\begin{aligned}&A(X^1\,\oplus \,\omega ^1,X^2\,\oplus \,\omega ^2)\\ {}&\quad =a[X^1,X^2]\,\oplus \, (b(\mathcal {L}_{X^1}\omega ^2-\mathcal {L}_{X^2}\omega ^1)+c\mathrm {d}\left<X^2\,\oplus \,\omega ^1,X^1\,\oplus \,\omega ^2\right>_-) \end{aligned}$$

for (uniquely determined by A) real numbers a, b, c.

Roughly speaking, Corollary 2.4 says that if $m\ge 2$, then any $\mathcal {M} f_m$-natural skew-symmetric bilinear operator A in the sense of Definition 2.1 coincides with the one given by Courant bracket $[-,-]^C$ up to three real constants.

Definition 2.5

A $\mathcal {M} f_m$-natural bilinear operator A in the sense of Definition 2.1 satisfies the Leibniz rule if

$$\begin{aligned} A(\rho _1,A(\rho _2,\rho _3))=A(A(\rho _1,\rho _2),\rho _3)+A(\rho _2,A(\rho _1,\rho _3)) \end{aligned}$$

for all $\rho _1,\rho _2,\rho _3\in \mathcal {X}(M)\,\oplus \,\Omega ^1(M)$ and all m-manifolds M.

Of course, in the case of skew-symmetric bilinear A the Leibniz rule is equivalent to the Jacobi identity $\sum _{{\mathrm {cycl}}(\rho _1,\rho _2,\rho _3)}A(\rho _1,A(A(\rho _2,\rho _3))=0$.

Example 2.6

The (not skew-symmetric) Courant bracket given by

$$\begin{aligned}&[X^1\oplus \omega ^1,X^2\oplus \omega ^2]_0\\&\quad : =[X^1,X^2]\,\oplus \,(\mathcal {L}_{X^1}\omega ^2-i_{X^2}\mathrm {d}\omega ^1), \end{aligned}$$

where $X^i\,\oplus \,\omega ^i \in \mathcal {X}(M)\,\oplus \,\Omega ^1(M)$, satisfies the Leibniz rule, see [7, 8].

The Courant bracket $[-,-]^C$ from Example 2.2 does not satisfy the Leibniz rule.

Theorem 2.7

[2]. If $m\ge 2$, any $\mathcal {M} f_m$-natural bilinear operator A in the sense of Definition 2.1 satisfying the Leibniz rule is one of the following ones:

$$\begin{aligned} A^{\left<1,a\right>}(\rho ^1,\rho ^2)= & {} a[X^1,X^2]\,\oplus \, 0, \\ A^{\left<2,a\right>}(\rho ^1,\rho ^2)= & {} a[X^1,X^2]\,\oplus \, (a(\mathcal {L}_{X^1}\omega ^2-\mathcal {L} _{X^2}\omega ^1)),\\ A^{\left<3,a\right>}(\rho ^1,\rho ^2)= & {} a[X^1,X^2]\,\oplus \, a\mathcal {L}_{X^1}\omega ^2, \\ A^{\left<4,a,0\right>}(\rho ^1,\rho ^2)= & {} a[X^1,X^2]\,\oplus \, (a(\mathcal {L}_{X^1}\omega ^2-i_{X^2}\mathrm{d}\omega ^1)), \end{aligned}$$

where a is an arbitrary real number, and where $\rho ^1=X^1\,\oplus \,\omega ^1$ and $\rho ^2=X^2\,\oplus \,\omega ^2$.

Corollary 2.8

If $m\ge 2$, the Courant bracket $[-,-]_0$ from Example 2.6 for m-manifolds M is the unique $\mathcal {M}f_m$-natural bilinear operator A in the sense of Definition 2.1 satisfying the conditions:

(A1):

$A(\rho _1, A(\rho _2,\rho _3))=A(A(\rho _1,\rho _2),\rho _3)+A(\rho _2,A(\rho _1,\rho _3)),$

(A2):

$\pi A(\rho _1,\rho _2)=[\pi \rho _1,\pi \rho _2],$

(A3):

$A(\rho _1,\rho _1)=i_0 \mathrm{d}\left<\rho _1,\rho _1\right>_+ ,$

for all $\rho _1,\rho _2,\rho _3\in \mathcal {X}(M)\,\oplus \,\Omega ^1(M)$ and all m-manifolds M, where $\left<-,-\right>_+$ is the pairing of Example 2.2, $\pi :TM\,\oplus \, T^*M\rightarrow TM$ is the fibred projection given by $\pi (v,\omega )=v$ and $i_0:T^*M\rightarrow TM\,\oplus \, T^*M$ is the fibred embedding $i_0(\omega )=(0,\omega )$.

Consequently, if $m\ge 2$, then a $\mathcal {M}f_m$-natural bilinear operator A in the sense of Definition 2.1 satisfying the conditions (A1)–(A3) satisfies the conditions:

(A4):

$\pi \rho _1\left<\rho _2,\rho _3\right>_+ =\left<A(\rho _1,\rho _2),\rho _3\right>_++\left<\rho _2,A(\rho _1,\rho _3\right>_+,$

(A5):

$A(\rho _1,f\rho _2)=\pi \rho _1(f)\rho _2+fA(\rho _1,\rho _2)$

for all $\rho _1,\rho _2\in \mathcal {X}(M)\,\oplus \,\Omega ^1(M)$, all $f\in \mathcal {C}^\infty (M)$ and all m-manifolds M (i.e., putting $[[-,-]]:=A$ we get an exact Courant algebroid $E=(TM\,\oplus \, T^*M, [[-,-]],$ $\left<-,-\right>_+, \pi , i_0)$ in the sense of [8] for any m-manifold M).

Proof

By Theorem 2.7, the conditions (A1) and (A2) imply that $A=A^{\left<1,1\right>}$ or $A=A^{\left<2,1\right>}$ or $A=A^{\left<3,1\right>}$ or $A=A^{\left<4,1,0\right>}$. On the other hand if $\rho _1=X\,\oplus \,\omega$, then $i_0\mathrm {d}\left<\rho _1,\rho _1\right>_+=0\,\oplus \, \mathrm {d}i_X\omega$ and $A^{\left<1,1\right>}(\rho _1,\rho _1)=0\,\oplus \, 0$ and $A^{\left<2,1\right>}(\rho _1,\rho _1)=0\,\oplus \, 0$ and $A^{\left<3,1\right>}(\rho _1,\rho _1)=0\,\oplus \,\mathcal {L}_X\omega$ and $A^{\left<4,1,0\right>}(\rho _1,\rho _1)=0\,\oplus \, \mathrm {d}i_X\omega$. Then $A=A^{\left<4,1,0\right>}$. $\square$

Corollary 2.9

If $m\ge 2$, any $\mathcal {M} f_m$-natural Lie algebra brackets on $\mathcal {X}(M)\,\oplus \,\Omega ^1(M)$ [i.e., $\mathcal {M}f_m$-natural skew-symmetric bilinear operator satisfying the Jacobi identity (Leibniz rule) is the constant multiple of the one of the following two Lie algebra brackets:

$$\begin{aligned}&[[X^1\,\oplus \, \omega ^1,X^2\,\oplus \,\omega ^2]]_1=[X^1,X^2]\,\oplus \, 0,\\ {}&[[X^1\,\oplus \,\omega ^1,X^2\,\oplus \,\omega ^2]]_2=[X^1,X^2]\,\oplus \, (\mathcal {L}_{X^1}\omega ^2-\mathcal {L}_{X^2}\omega ^1). \end{aligned}$$

At the end of this section we are going to describe completely all Lie algebroids $(TM\otimes T^*M, [[-,-]],a)$ which are invariant with respect to immersions ($\mathcal {M} f_m$-maps). The concept of Lie algebroids can be found in the fundamental book [6].

Of course, the anchor $a:TM\,\oplus \, T^*M\rightarrow TM$ for all m-manifolds M must be $\mathcal {M} f_m$-natural transformation [i.e., $Tf\circ a=a\circ (Tf\,\oplus \, T^*f)$ for any $\mathcal {M} f_m$-map $f:M\rightarrow M^1$] and fibre linear. By Corollary 2.9, $[[-,-]]=\mu [[-,-]]_1$ or $[[-,-]]=\mu [[-,-]]_2$ for some $\mu \in \mathbf {R}$.

Lemma 2.10

Any $\mathcal {M} f_m$-natural transformation $a:TM\,\oplus \, T^*M\rightarrow TM$ which is fibre linear is the constant multiple of the fibre projection $\pi :TM\,\oplus \, T^*M\rightarrow TM$.

Proof

Clearly, a is determined by the values $<\eta , a_x(v,\omega )>\in \mathbf {R}$ for all $\omega ,\eta \in T^*_xM$, $v\in T_xM$, $x\in M$, $M\in {\mathrm {Obj}}(\mathcal {M} f_m)$. By the standard chart arguments, we may assume $M=\mathbf {R}^m$, $x=0\ ,$ $\eta =\mathrm {d}_0x^1$. We can write $<\mathrm {d}_0x^1,a_0(v,\omega )>=\sum _i\alpha _iv^i+\sum _j\beta ^j\omega _j$, where $v^i$ are the coordinates of v and $\omega _j$ are the coordinates of $\omega$, and where $\alpha _i$ and $\beta ^j$ are the real numbers determined by $a_0$. Then using the invariance of $a_0$ with respect to the maps $(\tau ^1x^1,...,\tau ^mx^m)$ for $\tau ^1>0,...,\tau ^m>0$ we deduce that $\alpha _2=\cdots =\alpha _m=0$ and $\beta _1=\cdots =\beta _m=0$. Then the vector space of all a in question is at most 1-dimensional. Thus the dimension argument completes the proof. $\square$

So, $a=k\pi$ for some real number k. It must be $a([[X^1\,\oplus \, 0,X^2\,\oplus \, 0]])=[a(X^1\,\oplus \,0),a(X^2\,\oplus \, 0)]$ for any vector fields $X^1$ and $X^2$ on M. This gives the condition $k\mu [X^1,X^2]=k^2[X^1,X^2].$ Then $k\mu =k^2$, and then ($k=0$ and $\mu$ arbitrary) or ($k\not =0$ and $\mu =k$). Consider two cases:

1.  $[[-,-]]=\mu [[-,-]]_1$. Let $\rho ^1=X^1\,\oplus \,\omega ^1$ and $\rho ^2=X^2\,\oplus \,\omega ^2$. It must be $[[\rho ^1,f\rho ^2]]=a(\rho ^1)(f)\rho ^2+f[[\rho ^1,\rho ^2]]$. Considering the $\Omega ^1(M)$-parts of both sides of this equality we get $0=kX^1(f)\omega ^2+0$ for any vector fields $X^1,X^2$ on M any map $f:M\rightarrow \mathbf {R}$ and any $\omega ^1,\omega ^2\in \Omega ^1(M)$. Then $k=0$. Then considering the $\mathcal {X}(M)$-parts we get $\mu [X^1,fX^2]= f\mu [X^1,X^2]$. Then $\mu X^1(f)X^2=0$ for all vector fields $X^1$ and $X^2$ on M and all maps $f:M\rightarrow \mathbf {R}$, i.e., $\mu =0$.

2.  $[[-,-]]=\mu [[-,-]]_2$. Let $\rho ^1=0\,\oplus \, \omega ^1$ and $\rho ^2=X^2\,\oplus \, 0$. It must be $[[\rho ^1,f\rho ^2]]=a(\rho ^1)(f)\rho ^2+f[[\rho ^1,\rho ^2]]$. Considering the $\Omega ^1(M)$-parts of both sides of this equality we get $-\mu \mathcal {L}_{fX^2}\omega ^1=-\mu f\mathcal {L}_{X^2}\omega ^1$. Then $\mu =0$ or $\mathrm{d}i_{fX^2}\omega ^1+ i_{fX^2}\mathrm{d}\omega ^1= f\mathrm{d}i_{X^2}\omega ^1+ f i_{X^2}\mathrm{d}\omega ^1$. Putting $\omega ^1=\mathrm{d}g$ we get $\mu =0$ or $\mathrm{d}(i_{fX^2}\mathrm{d}g)=f\mathrm{d}i_{X^2}\mathrm{d}g$. Then $\mu =0$ or $\mathrm{d}(fX^2g)=f\mathrm{d}(X^2g)$. Then $\mu =0$ or $X^2(g)\mathrm{d}f=0$ for any $X^2,g,f$ in question. Putting $X^2={\partial \over \partial x^1}$ and $f=g=x^1$ we get $\mu =0$ or $\mathrm{d}x^1=0$. Then $\mu =0$, and then $k=\mu =0$.

On the other hand one can directly show that $(TM\,\oplus \, T^*M, 0[[-,-]]_1,0\pi )$ is a Lie algebroid. Thus we have

Proposition 2.11

If $m\ge 2$, $(TM\otimes T^*M,0,0)$ is the only invariant with respect to $\mathcal {M} f_m$-maps Lie algebroid $(EM,[[-,-]],a)$ with $EM=TM\,\oplus \, T^*M$.

3 The Natural Operators Similar to the Twisted Courant Bracket

Definition 3.1

A $\mathcal {M} f_m$-natural operator A sending p-forms $H\in \Omega ^p(M)$ on m-manifolds M into bilinear operators

$$\begin{aligned}A_H:({\mathcal {X}}(M)\,\oplus \, \Omega ^1(M))\times ({\mathcal {X}}(M)\,\oplus \,\Omega ^1(M))\rightarrow {\mathcal {X}}(M)\,\oplus \,\Omega ^1(M),\end{aligned}$$

is a $\mathcal {M} f_m$-invariant family of regular operators (functions)

$$\begin{aligned}A:\Omega ^p(M)\rightarrow Lin_2((\mathcal {X}(M)\,\oplus \,\Omega ^1(M))\times (\mathcal {X}(M)\,\oplus \,\Omega ^1(M)), \mathcal {X}(M)\,\oplus \,\Omega ^1(M))\end{aligned}$$

for all m-manifolds M, where $Lin_2(U\times V,W)$ denotes the vector space of all bilinear (over $\mathbf {R}$) functions $U\times V\rightarrow W$ for any real vector spaces U, V, W.

The $\mathcal {M} f_m$-invariance of A means that if $H^1\in \Omega ^p(M)$ and $H^2\in \Omega ^p(\overline{M})$ are $\varphi$-related and $(X^1\,\oplus \,\omega ^1, X^2\,\oplus \,\omega ^2)$ and $(\overline{X}^1\,\oplus \,\overline{\omega }^1,\overline{X}^2\,\oplus \,\overline{\omega }^2)$ are $\varphi$-related by an $\mathcal {M} f_m$-map $\varphi :M\rightarrow \overline{M}$, then so are $A_{H^1}(X^1\,\oplus \,\omega ^1,X^2\,\oplus \,\omega ^2)$ and $A_{H^2}(\overline{X}^1\,\oplus \,\overline{\omega }^1,\overline{X}^2\,\oplus \,\overline{\omega }^2)$.

The regularity of A means that it transforms smoothly parametrized families $(H_t,X^1_t\,\oplus \, \omega ^1_t,X^2_t\,\oplus \,\omega ^2_t)$ into smoothly parametrized families $A_{H_t}(X^1_t\,\oplus \,\omega ^1_t, X^2_t\,\oplus \,\omega ^2_t)$.

Example 3.2

The most important example of $\mathcal {M} f_m$-natural operator in the sense of Definition 3.1 for $p=3$ is given by the H-twisted Courant bracket

$$\begin{aligned}&[X^1\oplus \omega ^1,X^2\oplus \omega ^2]_H := [X^1,X^2]\,\oplus \,\left( \mathcal {L}_{X^1}\omega ^2-i_{X^2}\mathrm {d}\omega ^1+i_{X^1}i_{X^2}H\right) \ \end{aligned}$$

for all 3-forms $H\in \Omega ^3(M)$ and all m-manifolds M. We call this $\mathcal {M} f_m$-natural operator the twisted Courant bracket $\mathcal {M} f_m$-natural operator.

Example 3.3

The operator given by $[-,-]_{\mathrm{d}H}\$ for all $H\in \Omega ^2(M)$ and all m-manifolds M is a $\mathcal {M} f_m$-natural operator in the sense of Definition 3.1 for $p=2$.

The main result of this section is the following

Theorem 3.4

Assume $m\ge 3$. Then we have:

• 1. Any $\mathcal {M} f_m$-natural operator A in the sense of Definition 3.1 for $p=2$ such that $A_{H}=A_{H+\mathrm{d}H^1}$ for any $H\in \Omega ^2(M)$ and any $H^1\in \Omega ^1(M)$ is of the form

  $$\begin{aligned}&A_H (\rho ^1,\rho ^2) = a[X^1,X^2] \\&\quad \oplus \, \left( b_1\mathcal {L}_{X^2}\omega ^1+b_2\mathcal {L}_{X^1}\omega ^2 +b_3\mathrm {d}\left<\rho ^1,\rho ^2\right>_+ + b_4\mathrm {d}\left<\rho ^1,\rho ^2\right>_-+ ci_{X^1}i_{X^2}\mathrm {d}H\right) , \end{aligned}$$

  for (uniquely determined by A) reals $a, b_1,...,c$, where 2-forms $H\in \Omega ^2(M)$, pairs $\rho ^i=X^i\,\oplus \,\omega ^i\in \mathcal {X}(M)\,\oplus \,\Omega ^1(M)$ for $i=1,2$ and m-manifolds M are arbitrary.

• 2. Any $\mathcal {M} f_m$-natural operator (not necessarily satisfying $A_H=A_{H+\mathrm {d}H^1}$) in the sense of Definition 3.1 for $p=3$ is of the form

  $$\begin{aligned}&A_H (\rho ^1,\rho ^2) = a[X^1,X^2]\\ {}&\quad \oplus \, \left( b_1\mathcal {L}_{X^2}\omega ^1+b_2\mathcal {L}_{X^1}\omega ^2 +b_3\mathrm {d}\left<\rho ^1,\rho ^2\right>_+ + b_4\mathrm {d}\left<\rho ^1,\rho ^2\right>_-+ci_{X^1}i_{X^2}H\right) , \end{aligned}$$

  for (uniquely determined by A) reals $a, b_1,...,c$, where 3-forms $H\in \Omega ^3(M)$, pairs $\rho ^i=X^i\,\oplus \,\omega ^i\in \mathcal {X}(M)\,\oplus \,\Omega ^1(M)$ for $i=1,2$ and m-manifolds M are arbitrary.

• 3. If $p\ge 4$, any $\mathcal {M} f_m$-natural operator (not necessarily satisfying $A_H=A_{H+\mathrm {d}H^1}$) in the sense of Definition 3.1 is of the form

  $$\begin{aligned}&A_H (\rho ^1,\rho ^2) = a[X^1,X^2]\,\oplus \, \left( b_1\mathcal {L}_{X^2}\omega ^1+b_2\mathcal {L}_{X^1}\omega ^2 +b_3\mathrm {d}\left<\rho ^1,\rho ^2\right>_+ + b_4\mathrm {d}\left<\rho ^1,\rho ^2\right>_-\right) \end{aligned}$$

  for (uniquely determined by A) reals $a, b_1,...,b_4$, where p-forms $H\in \Omega ^p(M)$, pairs $\rho ^i=X^i\,\oplus \,\omega ^i\in \mathcal {X}(M)\,\oplus \,\Omega ^1(M)$ for $i=1,2$ and m-manifolds M are arbitrary.

Proof

Clearly, $A_0$, where 0 is the zero p-form, can be treated as the bilinear operator in the sense of Definition 2.1. Then $A_0$ is described in Theorem 2.3. So we can replace A by $A-A_0$. In other words, we have assumption $A_0=0$.

By the invariance, A is determined by the values $A_H(X^1\,\oplus \,\omega ^1,X^2\,\oplus \,\omega ^2)_{\vert _0}$ for all $H\in \Omega ^p(\mathbf {R}^m), X^i\,\oplus \,\omega ^i\in \mathcal {X}(\mathbf {R}^m)\oplus \Omega ^1(\mathbf {R}^m)$. Put

$$\begin{aligned}&A_H(X^1\,\oplus \,\omega ^1,X^2\,\oplus \,\omega ^2)_{\vert 0}\\&\quad =\left( A^1_H(X^1\,\oplus \,\omega ^1,X^2\,\oplus \,\omega ^2)_{\vert 0}, A^2_H(X^1\,\oplus \,\omega ^1,X^2\,\oplus \,\omega ^2)_{\vert 0}\right) ,\end{aligned}$$

where $A^1_H(...)_{\vert 0}\in T_0\mathbf {R}^m$ and $A^2_H(...)_{\vert 0}\in T^*\mathbf {R}^m$. Then A is determined by

$$\begin{aligned} \left<A^1_H(X^1\,\oplus \,\omega ^1,X^2\,\oplus \,\omega ^2)_{\vert _0},\eta \right>\in \mathbf {R}\; \text{ and }\; \left<A^2_H(X^1\,\oplus \,\omega ^1,X^2\,\oplus \,\omega ^2)_{\vert 0},\mu \right>\in \mathbf {R} \end{aligned}$$

for all $H\in \Omega ^p(\mathbf {R}^m) , X^i\,\oplus \,\omega ^i\in \mathcal {X}(\mathbf {R}^m)\oplus \Omega ^1(\mathbf {R}^m)$, $\eta \in T^*_0\mathbf {R}^m, \mu \in T_0\mathbf {R}^m$, $i=1,2$.

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 $(j^r_xH=j^r_x\overline{H} , j^r_x(\rho ^i)=j^r_x(\overline{\rho }^i), i=1,2)$ it follows $A_{H}(\rho ^1,\rho ^2)_{\vert x}=A_{\overline{H}}(\overline{\rho }^1,\overline{\rho }^2)_{\vert x}$. So, we may assume that $H, X^1,X^2, \omega ^1, \omega ^2$ are polynomials of degree not more than r.

Using the invariance of A with respect to the homotheties and the bi-linearity of $A_H$ (for given H) we obtain homogeneity condition

$$\begin{aligned}&\left\langle A^1_{\left( {1\over t}\mathrm {id}\right) _*H}\left( t\left( {1\over t}\mathrm {id}\right) _*X^1\,\oplus \, t\left( {1\over t}\mathrm {id}\right) _*\omega ^1, t\left( {1\over t}\mathrm {id}\right) _*X^2\,\right. \right. \\ {}&\qquad \left. \left. \oplus \, t\left( {1\over t}\mathrm {id}\right) _*\omega ^2\right) _{\vert _0},\eta \right\rangle \\ {}&\quad =t\left<A^1_H(X^1\,\oplus \,\omega ^1, X^2\,\oplus \,\omega ^2)_{\vert _0},\eta \right>. \end{aligned}$$

Then, by the homogeneous function theorem, since A is of finite order and regular and $A_0=0$ and $p\ge 2$, we have $\left<A^1_H(X^1\,\oplus \,\omega ^1,X^2\,\oplus \,\omega ^2)_{\vert _0},\eta \right>=0.$

Using the same arguments we get homogeneity condition

$$\begin{aligned}&\left\langle A^2_{\left( {1\over t}\mathrm {id}\right) _*H}\left( t\left( {1\over t}\mathrm {id}\right) _*X^1\oplus t\left( {1\over t}\mathrm {id}\right) _*\omega ^1, t\left( {1\over t}\mathrm {id}\right) _*X^2\right. \right. \\ {}&\qquad \left. \left. \oplus t\left( {1\over t}\mathrm {id}\right) _*\omega ^2\right) _{\vert _0},\mu \right\rangle \\ {}&\quad =t^3\left<A^2_H(X^1\oplus \omega ^1, X^2\oplus \omega ^2)_{\vert _0},\mu \right>. \end{aligned}$$

Then, if $p=2$, by the homogeneous function theorem and the bi-linearity of $A_H$ and the assumptions $A_0=0$ and $A_H=A_{H+\mathrm {d}H^1}$, the value $\left<A^2_H(X^1\oplus \omega ^1,X^2\oplus \omega ^2)_{\vert 0},\mu \right>$ depends quadrilinearly on $X^1{}_{\vert 0}$, $X^2{}_{\vert 0}$, $j^1_0(H-H_{\vert 0})$ and $\mu$, only. By $m\ge 3$ and the regularity of A, we may assume that $X^1_{\vert 0}$, $X^2_{\vert 0}$ and $\mu$ are linearly independent. Then by the invariance we may assume $X^1_{\vert 0}=\partial _1{}_{\vert 0}$, $X^2_{\vert 0}=\partial _2{}_{\vert 0}$ and $\mu =\partial _3{}_{\vert 0}$. Then A is determined by the values $\left<A^2_{x^{i_1}\mathrm{d}x^{i_2}\wedge \mathrm{d}x^{i_3}}(\partial _1\oplus 0,\partial _2\oplus 0),\partial _3{}_{\vert 0}\right>$ for all $i_1=1,...,m$ and $i_2,i_3$ with $1\le i_2<i_3\le m$. Then using the invariance of A with respect to $\tau \mathrm {id}$ for $\tau ^i>0$ we deduce that only $v:=\left<A^2_{x^1 \mathrm {d}x^2\wedge \mathrm {d}x^3}(\partial _1\oplus 0,\partial _2\oplus 0),\partial _{3}{}_{\vert 0}\right>$,$w:=\left<A^2_{x^2 \mathrm {d}x^1\wedge \mathrm {d}x^3}(\partial _1\oplus 0,\partial _2\oplus 0),\partial _{3}{}_{\vert 0}\right>$, $z:=\left<A^2_{x^3 \mathrm {d}x^1\wedge \mathrm {d}x^2}(\partial _1\oplus 0,\partial _2\oplus 0),\partial _{3}{}_{\vert 0}\right>$ may be not-zero. But $x^1\mathrm {d}x^2\wedge \mathrm {d}x^3=-x^2\mathrm {d}x^1\wedge \mathrm {d}x^3+\mathrm {d}(...)$. So, $v=-w$. Similarly, $v=-z$. Therefore the vector space of all A in question with $A_0=0$ and $A_{H}=A_{H+\mathrm {d}H^1}$ is at most one-dimensional. The part (1) of the theorem is complete. If $p=3$, then (by almost the same arguments as for $p=2$) A is determined by the values $\left<A^2_{\mathrm {d}x^{i_1}\wedge \mathrm {d}x^{i_2}\wedge \mathrm {d}x^{i_3}}(\partial _1\oplus 0,\partial _2\oplus 0),\partial _3{}_{\vert 0}\right>\in \mathbf {R}$ for all $i_1,i_2,i_3$ with $1\le i_1<i_2<i_3\le m$. Then using the invariance with respect to $(\tau ^1x^1,...\tau ^mx^m)$ for $\tau ^i>0$ we deduce that only the value $\left<A^2_{\mathrm{d}x^1\wedge \mathrm{d}x^2\wedge \mathrm{d}x^3}(\partial _1\oplus 0,\partial _2\oplus 0),\partial _{3}{}_{\vert 0}\right>\in \mathbb {R}$ 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 $0\oplus i_{X^1}i_{X^2}H$).

If $p\ge 4$, then (similarly as for $p=2$) $<A^2_H(X^1\oplus \omega ^1,X^2\oplus \omega ^2)_{\vert 0},\mu >=0$.

Theorem 3.4 is complete. $\square$

Corollary 3.5

If $m\ge 3$, any $\mathcal {M} f_m$-natural operator A in the sense of Definition 3.1 for $p=3$ such that $A_H$ is skew-symmetric for any $H\in \Omega ^3(M)$ and any m-manifold M is of the form

$$\begin{aligned}&A_H(X^1\oplus \omega ^1,X^2\oplus \omega ^2)=a[X^1,X^2]\\&\quad \oplus \left( b(\mathcal {L}_{X^1}\omega ^2-\mathcal {L}_{X^2}\omega ^1)+c\mathrm{d}\left<X^2\oplus \omega ^1,X^1\oplus \omega ^2\right>_-+ei_{X^1}i_{X^2}H\right) \end{aligned}$$

for (uniquely determined by A) real numbers a, b, c, e.

Roughly speaking, Corollary 3.5 says that any $\mathcal {M} f_m$-natural operator A in the sense of Definition 3.1 such that $A_H$ is skew-symmetric for any $H\in \Omega ^3(M)$ and any m-manifold M coincides with the “skew-symmetrization” of the twisted Courant bracket $\mathcal {M} f_m$-natural operator up to four real constants a, b, c, e.

Corollary 3.6

If $m\ge 3$, then the twisted Courant bracket $\mathcal {M} f_m$-natural operator from Example 3.2 is the unique $\mathcal {M} f_m$-natural operator A in the sense of Definition 3.1 for $p=3$ satisfying the following properties:

(B1):

$A_0(\rho _1,\rho _2)=[\rho _1,\rho _2]_0,$

(B2):

$A_H(X\oplus 0,Y\oplus 0)=[X,Y]\oplus i_{X}i_{Y}H$

for all closed $H\in \Omega _{cl}^3(M)$, all $\rho _1,\rho _2, X\oplus 0, Y\oplus 0\in \mathcal {X}(M)\oplus \Omega ^1(M)$ and all m-manifolds M, where $[-,-]_0$ is the $\mathcal {M} f_m$-natural bilinear operator given by the (not skew-symmetric) Courant bracket as in Example 2.6.

Proof

Clearly, the twisted Courant bracket $\mathcal {M} f_m$-natural operator satisfies (B1) and (B2). Consider A in question satisfying (B1) and (B2). Then by Theorem 3.4, there exist uniquely determined reals $a,b_1,...,c$ such that for all $H\in \Omega ^3(M)$ and m-manifolds M

$$\begin{aligned}&A_H (\rho ^1,\rho ^2) = a[X^1,X^2]\\ {}&\quad \oplus \left( b_1\mathcal {L}_{X^2}\omega ^1+b_2\mathcal {L}_{X^1}\omega ^2 +b_3\mathrm {d}\left<\rho ^1,\rho ^2\right>_+ + b_4\mathrm {d}\left<\rho ^1,\rho ^2\right>_-+ci_{X^1}i_{X^2}H\right) , \end{aligned}$$

where $\rho ^i=X^i\oplus \omega ^i\in \mathcal {X}(M)\oplus \Omega ^1(M)$ are arbitrary. Putting $\omega ^1=\omega ^2=0$ we get $A_H(\rho ^1,\rho ^2)=a[X^1,X^2]\oplus ci_{X^1}i_{X^2}H$. Then condition (B2) implies $c=1$. Putting $H=0$ we get

$$\begin{aligned}&A_0 (\rho ^1,\rho ^2)= a[X^1,X^2]\oplus \left( b_1\mathcal {L}_{X^2}\omega ^1+b_2\mathcal {L}_{X^1}\omega ^2 +b_3\mathrm {d}\left<\rho ^1,\rho ^2\right>_+ + b_4\mathrm {d}\left<\rho ^1,\rho ^2\right>_-\right) \end{aligned}$$

for all $\rho ^i=X^i\oplus \omega ^i\in \mathcal {X}(M)\oplus \Omega ^1(M)$ and all m-manifolds M. But $A_0$ is a $\mathcal {M} f_m$-natural bilinear operator in the sense of Definition 2.1. Then $a,b_1,b_2,b_3,b_4$ are uniquely determined because of Theorem 2.3. Then $a,b_1,...,c$ are uniquely determined. So, A is uniquely determined by conditions (B1) and (B2). $\square$

4 The Natural Operators Similar to the Twisted Courant Bracket and Defined for Closed p-Forms Only

In the previous section, we considered $\mathcal {M} f_m$-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

Definition 4.1

A $\mathcal {M} f_m$-natural operator A sending closed p-forms $H\in \Omega _{cl}^p(M)$ on m-manifolds M into bilinear operators

$$\begin{aligned} A_H:({\mathcal {X}}(M)\oplus \Omega ^1(M))\times ({\mathcal {X}}(M)\oplus \Omega ^1(M))\rightarrow {\mathcal {X}}(M)\oplus \Omega ^1(M),\end{aligned}$$

is a $\mathcal {M} f_m$-invariant family of regular operators (functions)

$$\begin{aligned}A:\Omega _{cl}^p(M)\!\rightarrow \! Lin_2((\mathcal {X}(M)\oplus \Omega ^1(M))\!\times \! (\mathcal {X}(M)\oplus \Omega ^1(M)), \mathcal {X}(M)\oplus \Omega ^1(M)),\end{aligned}$$

for all m-manifolds M.

We have the following corollary of Theorem 3.4.

Corollary 4.2

Assume $m\ge 3$. Then we have:

• 1. If $p=3$, any $\mathcal {M} f_m$-natural operator in the sense of Definition 4.1 is of the form

  $$\begin{aligned}&A_H (\rho ^1,\rho ^2) = a[X^1,X^2] \\ {}&\quad \oplus (b_1\mathcal {L}_{X^2}\omega ^1+b_2\mathcal {L}_{X^1}\omega ^2 +b_3\mathrm {d}\left<\rho ^1,\rho ^2\right>_+ + b_4\mathrm {d}\left<\rho ^1,\rho ^2\right>_-+ci_{X^1}i_{X^2}H), \end{aligned}$$

  for uniquely determined by A reals $a, b_1,...,c$, where closed 3-forms $H\in \Omega ^3_{cl}(M)$, pairs $\rho ^i=X^i\oplus \omega ^i\in \mathcal {X}(M)\oplus \Omega ^1(M)$ for $i=1,2$ and m-manifolds M are arbitrary.

• 2. If $p\ge 4$, any $\mathcal {M} f_m$-natural operator in the sense of Definition 4.1 is of the form

  $$\begin{aligned}&A_H (\rho ^1,\rho ^2) = a[X^1,X^2]\\ {}&\quad \oplus \left( b_1\mathcal {L}_{X^2}\omega ^1+b_2\mathcal {L}_{X^1}\omega ^2 +b_3\mathrm {d}\left<\rho ^1,\rho ^2\right>_+ + b_4\mathrm {d}\left<\rho ^1,\rho ^2\right>_-\right) \end{aligned}$$

  for uniquely determined by A reals $a, b_1,...,b_4$, where closed p-forms $H\in \Omega ^p_{cl}(M)$, pairs $\rho ^i=X^i\oplus \omega ^i$ for $i=1,2$ and m-manifolds M are arbitrary.

Proof

Let A be a $\mathcal {M} f_m$-natural operator in the sense of Definition 4.1 for p. Define a $\mathcal {M} f_m$-natural operator $A^1$ in the sense of Definition 3.1 for $p-1$ by $A^1_{{\tilde{H}}}=A_{\mathrm {d}{\tilde{H}}}.$ Then $A^1_{{\tilde{H}}+\mathrm {d}H_1}=A^1_{{\tilde{H}}}$ for any ${\tilde{H}}\in \Omega ^{p-1}(M)$ and $H_1\in \Omega ^{p-2}(M)$.

If $p=3$, then by Theorem 3.4, $A^1$ is of the form

$$\begin{aligned}&A^1_{{\tilde{H}}} (\rho ^1,\rho ^2) = a[X^1,X^2]\\ {}&\quad \oplus \left( b_1\mathcal {L}_{X^2}\omega ^1+b_2\mathcal {L}_{X^1}\omega ^2 +b_3\mathrm {d}\left<\rho ^1,\rho ^2\right>_+ + b_4\mathrm {d}\left<\rho ^1,\rho ^2\right>_-+ ci_{X^1}i_{X^2}\mathrm {d}{\tilde{H}}\right) \end{aligned}$$

for uniquely determined reals $a, b_1,...,c$ and all ${\tilde{H}}\in \Omega ^2(M)$, where $\rho ^i=X^i\oplus \omega ^i$ for $i=1,2$. Then

$$\begin{aligned}&A_H (\rho ^1,\rho ^2) = a[X^1,X^2]\\ {}&\quad \oplus \left( b_1\mathcal {L}_{X^2}\omega ^1+b_2\mathcal {L}_{X^1}\omega ^2 +b_3\mathrm {d}\left<\rho ^1,\rho ^2\right>_+ + b_4\mathrm {d}\left<\rho ^1,\rho ^2\right>_-+ci_{X^1}i_{X^2}H\right) \end{aligned}$$

for all exact 3-forms H, where $\rho ^i=X^i\oplus \omega ^i$ for $i=1,2$. But by the locality of A and the Poincare lemma we may replace the phrase “all exact 3-forms” by “all closed 3-forms”.

If $p\ge 4$, then by Theorem 3.4, $A^1$ is of the form

$$\begin{aligned}&A^1_{{\tilde{H}}} (\rho ^1,\rho ^2) = a[X^1,X^2]\\ {}&\quad \oplus \left( b_1\mathcal {L}_{X^2}\omega ^1+b_2\mathcal {L}_{X^1}\omega ^2 +b_3\mathrm {d}\left<\rho ^1,\rho ^2\right>_+ + b_4\mathrm {d}\left<\rho ^1,\rho ^2\right>_-+ ci_{X^1}i_{X^2}\mathrm {d}{\tilde{H}}\right) \end{aligned}$$

for uniquely determined reals $a, b_1,...,c$ (with arbitrary c if $p=4$ and with $c=0$ if $p\ge 5$) and all ${\tilde{H}}\in \Omega ^{p-1}(M)$, where $\rho ^i=X^i\oplus \omega ^i$ for $i=1,2$. The condition $A^1_{{\tilde{H}}}=A^1_{{\tilde{H}}+\mathrm {d}H_1}$ implies $ci_{X^1}i_{X^2}\mathrm {d}H_1=0$ for any $H_1\in \Omega ^{p-2}(M)$. If $p=4$, putting $X^1=\partial _1$, $X^2=\partial _2$ and $H_1=x^1\mathrm{d}x^2\wedge \mathrm{d}x^3$, we get $c(-\mathrm{d}x^3)=0$, i.e., $c=0$. If $p\ge 5$, then $c=0$, see above. Next, we proceed similarly as in the case $p=3$. $\square$

The above corollary and Theorem 3.4 imply

Theorem 4.3

If $m\ge 3$ and $p\ge 3$ then any $\mathcal {M} f_m$-natural operator in the sense of Definition 4.1 can be extended uniquely to a $\mathcal {M} f_m$-natural operator in the sense of Definition 3.1.

Roughly speaking, if $m\ge 3$ and $p\ge 3$, then any $\mathcal {M} f_m$-natural operator in the sense of Definition 4.1 can be treated as the $\mathcal {M} f_m$-natural operator in the sense of Definition 3.1, and vice-versa.

5 The Natural Operators Similar to the Twisted Courant Bracket and Satisfying the Leibniz Rule for Closed 3-Forms

Definition 5.1

A $\mathcal {M} f_m$-natural operator A in the sense of Definition 3.1 (or equivalently in the sense of Definition 4.1) satisfies the Leibniz rule for closed p-forms if

$$\begin{aligned}A_H(\rho _1,A_H(\rho _2,\rho _3))=A_H(A_H(\rho _1,\rho _2),\rho _3)+A_H(\rho _2,A_H(\rho _1,\rho _3)) \end{aligned}$$

for all $\rho _1,\rho _2,\rho _3\in \mathcal {X}(M)\oplus \Omega ^1(M)$, all closed p-forms $H\in \Omega _{cl}^p(M)$ and all m-manifolds M.

Example 5.2

The twisted Courant bracket $\mathcal {M} f_m$-natural operator presented in Example 3.2 satisfies the Leibniz rule for closed 3-forms, see [3, 8].

Theorem 5.3

If $m\ge 3$, any $\mathcal {M} f_m$-natural operator A in the sense of Definition 3.1 (or equivalently of Definition 4.1) for $p=3$ satisfying the Leibniz rule for closed 3-forms is one of the $\mathcal {M} f_m$-natural operators:

$$\begin{aligned} A^{\left<1,a\right>}_H(\rho _1,\rho _2)=&{} a[X^1,X^2]\oplus 0 ,\\ A^{\left<2,a\right>}_H(\rho ^1,\rho ^2)=&{} a[X^1,X^2]\oplus (a(\mathcal {L}_{X^1}\omega ^2-\mathcal {L} _{X^2}\omega ^1)),\\ A^{\left<3,a\right>}_H(\rho ^1,\rho ^2)=&{} a[X^1,X^2]\oplus (a\mathcal {L}_{X^1}\omega ^2), \\ A^{\left<4,a,e\right>}_H(\rho ^1,\rho ^2)=&{} a[X^1,X^2]\oplus (a(\mathcal {L}_{X^1}\omega ^2- i_{X^2}\mathrm {d}\omega ^1)+ei_{X^1}i_{X^2}H), \end{aligned}$$

where $\rho ^1=X^1\oplus \omega ^1$ and $\rho ^2=X^2\oplus \omega ^2$, and a and e are arbitrary real numbers.

Proof

Let A be a $\mathcal {M} f_m$-natural operator in the sense of Definition 3.1 for $p=3$ such that $A_H$ satisfies the Leibniz rule for any closed $H\in \Omega ^3_{cl}(M)$. By Theorem 3.4, A is of the form

$$\begin{aligned}&A_H(X^1\oplus \omega ^1,X^2\oplus \omega ^2) =a [X^1,X^2]\\ {}&\quad \oplus (b_1\mathcal {L}_{X^2}\omega ^1+b_2 \mathcal {L}_{X^1}\omega ^2+c_1 \mathrm {d}i_{X^2}\omega ^1+c_2 \mathrm {d}i_{X^1}\omega ^2+ei_{X^1}i_{X^2}H), \end{aligned}$$

for (uniquely determined by A) real numbers $a, b_1,b_2,c_1,c_2,e$. Then for any $X^1,X^2,$ $X^3\in \mathcal {X}(M)$ and $\omega ^1,\omega ^2,\omega ^3\in \Omega ^1(M)$ we have

$$\begin{aligned}&A_H(X^1\oplus \omega ^1, A_H(X^2\oplus \omega ^2,X^3\oplus \omega ^3))=a^2[X^1,[X^2,X^3]]\oplus \Omega ,\\&A_H(A_H(X^1\oplus \omega ^1,X^2\oplus \omega ^2),X^3\oplus \omega ^3)=a^2[[X^1,X^2],X^3]\oplus \Theta ,\\&A_H(X^2\oplus \omega ^2,A_H(X^1\oplus \omega ^1,X^3\oplus \omega ^3))=a^2[X^2,[X^1,X^3]]\oplus \mathcal {T}, \end{aligned}$$

where

$$\begin{aligned} \Omega= & {} b_1\mathcal {L}_{a[X^2,X^3]}\omega ^1+ c_1\mathrm{d}i_{a[X^2,X^3]}\omega ^1 + ei_{X^1}i_{a[X^2,X^3]}H\\&+\, b_2\mathcal {L}_{X^1}( b_1\mathcal {L}_{X^3}\omega ^2 +b_2\mathcal {L}_{X^2}\omega ^3+c_1\mathrm{d}i_{X^3}\omega ^2+c_2\mathrm{d}i_{X^2}\omega ^3+ei_{X^2}i_{X^3}H)\\&+\,c_2\mathrm{d}i_{X^1}( b_1\mathcal {L}_{X^3}\omega ^2+b_2\mathcal {L}_{X^2}\omega ^3+c_1\mathrm{d}i_{X^3}\omega ^2+c_2\mathrm{d}i_{X^2}\omega ^3+ei_{X^2}i_{X^3}H) ,\\ \Theta= & {} b_2\mathcal {L}_{a[X^1,X^2]}\omega ^3 + c_2\mathrm{d}i_{a[X^1,X^2]}\omega ^3+ei_{a[X^1,X^2]}i_{X^3}H\\&+\,b_1\mathcal {L}_{X^3}( b_1\mathcal {L}_{X^2}\omega ^1+ b_2\mathcal {L}_{X^1}\omega ^2+c_1\mathrm{d}i_{X^2}\omega ^1+ c_2\mathrm{d}i_{X^1}\omega ^2+e_{X^1}i_{X^2}H)\\&+\,c_1\mathrm{d}i_{X^3}(b_1\mathcal {L}_{X^2}\omega ^1+b_2\mathcal {L}_{X^1}\omega ^2+c_1\mathrm{d}i_{X^2}\omega ^1+c_2\mathrm{d}i_{X^1}\omega ^2+ei_{X^1}i_{X^2}H) ,\\ \mathcal {T}= & {} b_1\mathcal {L}_{a[X^1,X^3]}\omega ^2+c_1\mathrm{d}i_{a[X^1,X^3]}\omega ^2+ei_{X^2}i_{a[X^1,X^3]}H\\&+\,b_2\mathcal {L}_{X^2}(b_1\mathcal {L}_{X^3}\omega ^1+ b_2\mathcal {L}_{X^1}\omega ^3 +c_1\mathrm{d}i_{X^3}\omega ^1+c_2 \mathrm{d}i_{X^1}\omega ^3+ei_{X^1}i_{X^3}H)\\&+\,c_2 \mathrm{d}i_{X^2}(b_1\mathcal {L}_{X^3}\omega ^1 +b_2\mathcal {L}_{X^1}\omega ^3+c_1\mathrm{d}i_{X^3}\omega ^1+c_2\mathrm{d}i_{X^1}\omega ^3+ei_{X^1}i_{X^3}H). \end{aligned}$$

The Leibniz rule of $A_H$ is equivalent to $\Omega =\Theta +\mathcal {T}.$

Putting $H=0$, we are in the situation of Theorem 2.7. Then by Theorem 2.7 (i.e., by Theorem 3.2 in [2]) we get $(b_1,b_2,c_1,c_2)=(0,0,0,0)$ or $(b_1,b_2,c_1,c_2)=(0,a,0,0)$ or $(b_1,b_2,c_1,c_2)=(-a,a,0,0)$ or $(b_1,b_2,c_1,c_2)=(-a,a,a,0)$. More, $A_0$ for such $(b_1,b_2,c_1,c_2)$ satisfies the Leibniz rule.

Therefore (as $c_2=0$) the Leibniz rule of $A_H$ is equivalent to the equality

$$\begin{aligned}&eai_{X^1}i_{[X^2,X^3]}H + b_2e\mathcal {L}_{X^1} i_{X^2}i_{X^3}H\\&\quad = eai_{[X^1,X^2]}i_{X^3}H +b_1e\mathcal {L}_{X^3} i_{X^1}i_{X^2}H +c_1e\mathrm{d}i_{X^3} i_{X^1}i_{X^2}H\\&\qquad +eai_{X^2}i_{[X^1,X^3]}H+ b_2e\mathcal {L}_{X^2} i_{X^1}i_{X^3}H . \end{aligned}$$

If $(b_1,b_2,c_1,c_2)=(0,0,0,0)$, the above equality is equivalent to

$$\begin{aligned} eai_{X^1}i_{[X^2,X^3]}H=eai_{[X^1,X^2]}i_{X^3}H+ea i_{X^2}i_{[X^1,X^3]}H. \end{aligned}$$

Putting $X^1=\partial _1$, $X^2=\partial _1+x^1\partial _3$ and $X^3=\partial _2$ we have $[X^2,X^3]=0$, $[X^1,X^3]=0$ and $[X^1,X^2]=\partial _3$, and then $0=eai_{\partial _3}i_{\partial _2}H$ for any closed H (for example for $H=\mathrm{d}x^1\wedge \mathrm{d}x^2\wedge \mathrm{d}x^3$). Consequently $e=0$ or $a=0$.

If $(b_1, b_2,c_1,c_2)=(0,a,0,0)$, the above equality is equivalent to

$$\begin{aligned}&eai_{X^1}i_{[X^2,X^3]}H+ea \mathcal {L}_{X^1}i_{X^2}i_{X^3} H\\&\quad =eai_{[X^1,X^2]}i_{X^3}H+ea i_{X^2}i_{[X^1,X^3]}H+ea\mathcal {L}_{X^2}i_{X^1}i_{X^3}H. \end{aligned}$$

Putting $X^1=\partial _1$, $X^2=\partial _2$ and $X^3=\partial _3$ and $H=x^2\mathrm{d}x^1\wedge \mathrm{d}x^2\wedge \mathrm{d}x^3$ (it is closed) we have $[X^2,X^3]=0$, $[X^1,X^2]=0$, $[X^1,X^3]=0$, $\mathcal {L}_{X^2}i_{X^1}i_{X^3}H=\mathcal {L}_{\partial _2}x^2\mathrm{d}x^2=\mathrm{d}x^2$ and $\mathcal {L}_{X^1}i_{X^2}i_{X^3}H=\mathcal {L}_{\partial _1}(-x^2\mathrm{d}x^1)=0$. Then $ea\mathrm{d}x^2=0$. So, $a=0$ or $e=0$.

If $(b_1,b_2,c_1,c_2)=(-a,a,0,0)$, the above equality is equivalent to

$$\begin{aligned}&eai_{X^1}i_{[X^2,X^3]}H+ea \mathcal {L}_{X^1}i_{X^2}i_{X^3} H\\&\quad = eai_{[X^1,X^2]}i_{X^3}H-ea \mathcal {L}_{X^3}i_{X^1}i_{X^2}H +ea i_{X^2}i_{[X^1,X^3]}H+ea\mathcal {L}_{X^2}i_{X^1}i_{X^3}H. \end{aligned}$$

Putting $X^1=\partial _1$, $X^2=\partial _2$ and $X^3=\partial _3$ and $H=x^2\mathrm{d}x^1\wedge \mathrm{d}x^2\wedge \mathrm{d}x^3$ we have (see above) $[X^2,X^3]=0$, $[X^1,X^2]=0$, $[X^1,X^3]=0$, $\mathcal {L}_{X^2}i_{X^1}i_{X^3}H=\mathrm{d}x^2$, $\mathcal {L}_{X^1}i_{X^2}i_{X^3}H=0$ and $\mathcal {L}_{X^3}i_{X^1}i_{X^2}H=\mathcal {L}_{\partial _3}(-x^2\mathrm{d}x^3)=0$. Then $ea\mathrm{d}x^2=0$. So, $a=0$ or $e=0$.

If $(b_1,b_2,c_1,c_2)=(-a,a,a,0)$, the above equality is equivalent to

$$\begin{aligned}ea\sum \left\{ i_{X^1}i_{[X^2,X^3]}H+\mathcal {L}_{X^1}i_{X^2}i_{X^3}H\right\} =ea\mathrm{d}i_{X^1}i_{X^2}i_{X^3}H,\end{aligned}$$

where $\sum$ is the cyclic sum $\sum _{cycl(X^1,X^2,X^3)}$. Then e is arbitrary real number because from $\mathrm {d}H=0$ it follows

$$\begin{aligned}\sum \left\{ i_{X^1}i_{[X^2,X^3]}H+\mathcal {L}_{X^1}i_{X^2}i_{X^3}H\right\} =\mathrm{d}i_{X^1}i_{X^2}i_{X^3}H.\end{aligned}$$

Indeed, using $\mathrm {d}H=0$ and $i_{[X^1,X^4]}=\mathcal {L}_{X^1}i_{X^4}-i_{X^4}\mathcal {L}_{X^1}$ and the well-known formula expressing $\mathrm {d}H(X^1,X^2,X^3,X^4)$, we have

$$\begin{aligned}&\sum \left\{ i_{X^4}i_{X^1}i_{[X^2,X^3]}H+i_{X^4}\mathcal {L}_{X^1}i_{X^2}i_{X^3}H\right\} \\ {}&\quad = \sum \left\{ i_{X^4}i_{X^1}i_{[X^2,X^3]}H+\mathcal {L}_{X^1}i_{X^4}i_{X^2}i_{X^3}H-i_{[X^1,X^4]}i_{X^2}i_{X^3}H\right\} \\ {}&\quad = 6\sum \{H([X^2,X^3],X^1,X^4)+X^1H(X^3,X^2,X^4)\\ {}&\qquad -H(X^3,X^2,[X^1,X^4])\}\\ {}&\quad =-24\mathrm {d}H(X^1,X^2,X^3,X^4)+6X^4H(X^3,X^2,X^1)=i_{X^4}\mathrm {d}i_{X^1}i_{X^2}i_{X^3}H. \end{aligned}$$

Summing up, given a real number $a\not =0$ we have $(b_1,b_2,c_1,c_2,e)=(0,0,0,0,0)$ or $(b_1,b_2,c_1,c_2,e)=(0,a,0,0,0)$ or $(b_1,b_2,c_1,c_2,e)=(-a,a,0,0,0,)$ or $(b_1,b_2,c_1,c_2,e)=(-a,a,a,0,e)$, where e may be arbitrary real number. If $a=0$ we have $(b_1,b_2,c_1,c_2,e)=(0,0,0,0,e)$, where e may be arbitrary. Theorem 5.3 is complete. $\square$

Corollary 5.4

If $m\ge 3$, then the twisted Courant bracket $\mathcal {M} f_m$-natural operator from Example 3.2 is the unique $\mathcal {M} f_m$-natural operator A in the sense of Definition 3.1 for $p=3$ satisfying the following conditions:

(C1):

$A_H(\rho _1, A_H(\rho _2,\rho _3))=A_H(A_H(\rho _1,\rho _2),\rho _3)+A_H(\rho _2,A_H(\rho _1,\rho _3)),$

(C2):

$A_H(X\oplus 0,Y\oplus 0)=[X,Y]\oplus i_Xi_YH$

for all $\rho _1,\rho _2,\rho _3, X\oplus 0,Y\oplus 0\in \mathcal {X}(M)\oplus \Omega ^1(M)$, all closed $H\in \Omega _{cl}^3(M)$ and all m-manifolds M.

Proof

Indeed, the condition (C1) and Theorem 5.3 imply that $A=A^{\left<1,a\right>}$ or $A=A^{\left<2,a\right>}$ or $A=A^{\left<3,a\right>}$ or $A=A^{\left<4,a,e\right>}$ for some real numbers a and e. Then (C2) implies that $A=A^{\left<4,a,e\right>}$ and $a=1$ and $e=1$ because $A_H^{\left<1,a\right>}(X\oplus 0,Y\oplus 0)=a[X,Y]\oplus 0$ and $A_H^{\left<2,a\right>}(X\oplus 0,Y\oplus 0)=a[X,Y]\oplus 0$ and $A_H^{\left<3,a\right>}(X\oplus 0,Y\oplus 0)=a[X,Y]\oplus 0$ and $A^{\left<4,a,e\right>}_H(X\oplus 0,Y\oplus 0) =a[X,Y]\oplus ei_Xi_YH$. $\square$

Corollary 5.5

If $m\ge 3$, any $\mathcal {M} f_m$-natural operator A in the sense of Definition 3.1 for $p=3$ such that $A_H$ is a Lie algebra bracket (i.e., it is skew-symmetric, bilinear and satisfying the Leibniz rule) for all closed 3-forms $H\in \Omega ^3_{cl}(M)$ and all m-manifolds M is one of the $\mathcal {M} f_m$-natural operators:

$$\begin{aligned} A^{\left<1,a\right>}_H(\rho _1,\rho _2)=&{} a[X^1,X^2]\oplus 0 ,\\ A^{\left<2,a\right>}_H(\rho ^1,\rho ^2)=&{} a[X^1,X^2]\oplus (a(\mathcal {L}_{X^1}\omega ^2-\mathcal {L} _{X^2}\omega ^1)), \\ A^{\left<4,0,e\right>}_H(\rho ^1,\rho ^2)=&{} 0\oplus ei_{X^1}i_{X^2}H, \end{aligned}$$

where $\rho ^1=X^1\oplus \omega ^1$ and $\rho ^2=X^2\oplus \omega ^2$, and a and e are arbitrary real numbers.

Proof

It follows from Theorem 5.3. $\square$

Corollary 5.6

If $m\ge 3$, any $\mathcal {M} f_m$-natural operator A in the sense of Definition 3.1 for $p=3$ satisfying the Leibniz rule for all 3-forms H (or for all closed 3-forms and at least one non-closed 3-form) is one of the $\mathcal {M} f_m$-natural operators:

$$\begin{aligned} A^{\left<1,a\right>}_H(\rho _1,\rho _2)=&{} a[X^1,X^2]\oplus 0 ,\\ A^{\left<2,a\right>}_H(\rho ^1,\rho ^2)=&{} a[X^1,X^2]\oplus (a(\mathcal {L}_{X^1}\omega ^2-\mathcal {L} _{X^2}\omega ^1)),\\ A^{\left<3,a\right>}_H(\rho ^1,\rho ^2)=&{} a[X^1,X^2]\oplus (a\mathcal {L}_{X^1}\omega ^2), \\ A^{\left<4,a,0\right>}_H(\rho ^1,\rho ^2)=&{} a[X^1,X^2]\oplus (a(\mathcal {L}_{X^1}\omega ^2-i_{X^2}\mathrm {d}\omega ^1)),\\ A^{\left<4,0,e\right>}_H(\rho ^1,\rho ^2)=&{} 0\oplus ei_{X^1}i_{X^2}H,\end{aligned}$$

where $\rho ^1=X^1\oplus \omega ^1$ and $\rho ^2=X^2\oplus \omega ^2$, and a and e are arbitrary real numbers.

Proof

It follows from Theorem 5.3 and its proof. $\square$

Remark 5.7

It is well-known that given closed 3-form $H\in \Omega ^3_{cl}(M)$ on a m-manifold M, the twisted Courant bracket $[-,-]_H:(\mathcal {X}(M)\oplus \Omega ^1(M))\times (\mathcal {X}(M)\oplus \Omega ^1(M))\rightarrow \mathcal {X}(M)\oplus \Omega ^1(M)$ is bilinear and satisfies the properties (A1)–(A5) from Corollary 2.8 for all $\rho _1,\rho _2,\rho _3\in \mathcal {X}(M)\oplus \Omega ^1(M)$ and all $f\in \mathcal {C}^\infty (M)$, see [3, 8], but $[-,-]_H\not =[-,-]_0$ if $H\not =0$. Is it a contradiction with the uniqueness from Corollary 2.8? No, it is not. Indeed, $[-,-]_H$ is not extendable to a $\mathcal {M} f_m$-natural bilinear operator in the sense of Definition 2.1 because it is invariant only with respect to $\mathcal {M} f_m$-maps $\varphi :M\rightarrow M$ preserving H, in fact.

Remark 5.8

By Corollary 5.5, given a closed 3-form H on M, the skew-symmetric bracket $[[X^1\oplus \omega ^1,X^2\oplus \omega ^2]]^{(H)}: =0\oplus i_{X^1}i_{X^2}H$ satisfies the Leibniz rule. One can easily directly verify that $(TM\oplus T^*M,e[[-,-]]^{(H)},0\pi )$ for arbitrary fixed $e\in \mathbf {R}$ and closed 3-form H is a Lie algebroid canonically depending on H. So, if we have a closed 3-form H on a m-manifold M, we can construct canonical (in H) Lie algebroids $(EM, [[-,-]]^{[H]}, a^{[H]})$ with $EM=TM\oplus T^*M$ different than the one from Proposition 2.11.