On the Courant bracket on couples of vector fields and \(p\)-forms

If \(m\geq p+1\geq 2\) (or \(m=p\geq 3\)), all  natural bilinear  operators \(A\) transforming pairs of couples of vector fields and \(p\)-forms on \(m\)-manifolds \(M\) into couples of vector fields and \(p\)-forms on \(M\) are described. It is observed that  any natural skew-symmetric bilinear operator \(A\) as above coincides with the generalized Courant bracket up to three (two, respectively) real constants.


Introduction.
In the whole paper the word "bilinear" means "bilinear over R".
Let Mf m be the category of m-dimensional C ∞ manifolds and their embeddings.
The "doubled" tangent bundle T ⊕T * over Mf m is full of interest because of the non-degenerate symmetric bilinear form and the Courant bracket, see [2].The non-degenerate symmetric bilinear form and the Courant bracket on T ⊕ T * are involved in the definitions of Dirac and generalized complex structures, see e.g.[2,6,7].Such structures have applications in the high energy physics, see e.g.[1].That is why, in [4], we studied brackets similar to the Courant one.
The Courant bracket can be generalized to the one on T ⊕ p T * , see e.g.[7].That is why, in the present note, we study all Mf m -natural bilinear transforming pairs of couples X i ⊕ ω i ∈ X (M ) ⊕ Ω p (M ) (i = 1, 2) of vector fields and p-forms on m-manifolds M into couples A(X 1 ⊕ ω 1 , X 2 ⊕ ω 2 ) ∈ X (M ) ⊕ Ω p (M ) of vector fields and p-forms on M .Roughly speaking, in the present note, we deduce that if m ≥ p+1 ≥ 2 (or m = p ≥ 3, respectively), then any Mf m -natural skew-symmetric bilinear operator A as above coincides with the generalized Courant bracket up to three (or two, respectively) real constants.
Some linear natural operators on vector fields, forms and some other tensor fields have been studied in many papers, see e.g.[3,5,9,10], etc.
From now on, (x i ) (i = 1, . . ., m) denote the usual coordinates on R m and ∂ i = ∂ ∂x i are the canonical vector fields on R m .

The basic notions. Definition 2.1. An Mf m -natural bilinear operator
for m-dimensional manifolds M , where X (M ) is the space of vector fields on M and Ω p (M ) is the space of p-forms on M .
Remark 2.2.We recall that the Mf m -invariance of A means that if and T is an Mf m -invariant family of bilinear operators for m-manifolds M .
Remark 2.5.By the multi-linear Peetre theorem, see [8], any Mf m -natural bilinear operator A (as above) is of finite order.It means that there is a finite number r such that we have the following implication Remark 2.6.We say that an operator A is regular if it transforms smoothly parametrized families of objects into smoothly parametrized families.One can show that Mf m -natural bilinear operators are regular because of the Peetre theorem.Definition 2.7.An Mf m -natural operator B : T ⊕ T (0,0) p T * is an Mf m -invariant family of regular (not necessarily bilinear) operators

Example 2.8 ([7]
).The generalized Courant bracket is given by for any , where d is the usual differentiation, L is the Lie derivative, i is the usual inner differentiation and [−, −] is the usual bracket on vector fields.For p = 1 we get the usual Courant bracket as in [2].
Remark 2.9.If m = p, we have L X ω = di X ω+i X dω = di X ω for any vector field X and any m-form ω on an m-manifold M as dω = 0. Consequently, if m = p the generalized Courant bracket satisfies

The main result.
The main result of the present note is the following classification theorem.
Proof.Theorem 3.1 is an immediate consequence of Propositions 3.2 and 3.4.
T is of the form for a (uniquely determined by A) real number a.
Proof.For p = 1, our Proposition 3.2 is exactly Proposition 4.1 in [4].(The proof of Proposition 4.1 in [4] works for m = 1, too.)Let A be a Mf m -natural bilinear operator in question.We define new Mf m -natural bilinear operator Ã : for any m-manifold M and any vector fields X 1 , X 2 on M .
Our operator A is determined by the values A(X Moreover, by the invariance and the regularity of A and the Frobenius theorem we may additionally assume that X 1 = ∂ 1 and η = η o = d 0 x 1 .In other words, A is determined by the values Using the invariance of A with respect to the homotheties and the bilinearity of A, we have the homogeneity condition So, by the homogeneous function theorem, since A is of finite order and regular, the value A(∂ 1 ⊕ ω 1 , X ⊕ ω 2 ) | 0 , η o depends on j 1 0 X, only.Then A is determined by the values A(∂ 1 ⊕ 0, X ⊕ 0) | 0 , η o , i.e.A is determined by the number a.
Consequently, the vector space of all Mf m -natural bilinear operators T is not more than 1-dimensional.On the other hand, we have the Mf m -natural bilinear operator A o (in question) given by Proof.By the classical Peetre theorem (since B is linear in f ), B is of finite order in f , i.e. for any m-manifold M , any point x ∈ M and any vector field X ∈ X (M ) there is a natural number r such that for any by regularity of B, we may assume that X | 0 ∧ v = 0, and then by the invariance of B and the Frobenius theorem, we may assume that ) Since B is of finite order in f , we may assume that f is a polynomial.Now, by the invariance of B with respect to the diffeomorphisms (t 1 x 1 , . . ., t m x m ), t l ∈ R + , l = 1, . . ., m and the conditions of B, we derive that if , respectively) for all g = g(x 2 , . . ., x m ).By the invariance of B with respect to the diffeomorphisms of the form id R ×ψ we may assume that g = 1 or g = x 2 .Then if m ≥ p+1 ≥ 3 (or m = p ≥ 3, respectively), by the invariance of B with respect to the homotheties and the assumptions on B, we deduce B(∂ 1 , (x 1 ) 2 g) | 0 = 0 (or B(∂ 1 , (x 1 ) 3 g) | 0 = 0, respectively).Consequently, B = 0 if m ≥ p + 1 ≥ 3 (or m = p ≥ 3, respectively).The proof of Lemma 3.3 is complete.p T * for j = 1, . . ., 4 (or j = 1, 2) given by , Proof.Clearly, A is determined by the values Consequently, using the bilinearity of A, A is determined by the values 0 (or b1 = b2 = 0, respectively), i.e. we may assume that A is determined by the values d k (or dk , respectively), i.e. we may assume that A is determined by the value respectively) for all vector fields Y ∈ X (R m−1 ) (depending on x 2 , . . ., x m ).Next, by the regularity of A, we may assume that Y | 0 = 0, and then, by the invariance of A with respect to that local diffeomorphisms of the form id R ×ψ(x 2 , . . ., x m ) and the Frobenius theorem, we may assume that Y respectively) because of the invariance of A with respect to the homotheties.Consequently, A is determined by the Mf m -natural operator B : T ⊕ T (0,0) p T * given by B(X ⊕ f ) := A(X ⊕ 0, f X ⊕ 0), M ∈ obj(Mf m ), X ∈ X (M ), f ∈ C ∞ (M ).
Clearly, B satisfies the assumptions of Lemma 3.
for uniquely determined by A real numbers a, b, c (or a, b, c with c = 0, respectively), i.e. roughly speaking, any such A coincides with the generalized Courant bracket up to three (or two, respectively) real constants.