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.
Similar content being viewed by others
Avoid common mistakes on your manuscript.
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
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
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
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)
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
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
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
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
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
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
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:
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:
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
is a \(\mathcal {M} f_m\)-invariant family of regular operators (functions)
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
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
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
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
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
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
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
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
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
is a \(\mathcal {M} f_m\)-invariant family of regular operators (functions)
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
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
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
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
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:
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
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
where
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
If \((b_1,b_2,c_1,c_2)=(0,0,0,0)\), the above equality is equivalent to
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
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
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
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
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
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:
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:
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.
References
Courant, T.: Dirac manifolds. Trans. Am. Math. Soc. 319(631), 631–661 (1990)
Doupovec, M., Kurek, J., Mikulski, W.M.: The natural brackets on couples of vector fields and \(1\)-forms. Turk. J. Math. 42(2), 1853–1862 (2018)
Gualtieri, M.: Generalized complex geometry. Ann. Math. 174(1), 75–123 (2011)
Hitchin, N.: Generalized Calabi–Yau manifolds. Q. J. Math. 54(3), 281–308 (2003)
Kolář, I., Michor, P.W., Slovák, J.: Natural Operations in Differential Geometry. Springer, Berlin (1993)
Mackenzie, K.C.H.: General Theory of Lie Groupoids and Lie Algebroids. London Math. Soc., Lecture Note 213. Cambridge University Press, Cambridge (2005)
Liu, Z.J., Weinstein, A., Xu, P.: Main triples for Lie bialgebroids. J. Differ. Geom. 45, 547–574 (1997)
Ševera, P., Weinstein, A.: Poisson geometry with a \(3\)-form bacground. Progr. Teoret. Phys. Suppl., 145–154 (2001)
Acknowledgements
I would like to thank the reviewer for valuable suggestions. By one of them I was inspired to study the problem given in Proposition 2.11.
Author information
Authors and Affiliations
Corresponding author
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
About this article
Cite this article
Mikulski, W.M. The Natural Operators Similar to the Twisted Courant Bracket One. Mediterr. J. Math. 16, 101 (2019). https://doi.org/10.1007/s00009-019-1367-1
Received:
Revised:
Accepted:
Published:
DOI: https://doi.org/10.1007/s00009-019-1367-1