Abstract
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.
Highlights
In the whole paper the word “bilinear” means “bilinear over R”.Let Mfm be the category of m-dimensional C∞ manifolds and their embeddings.The “doubled” tangent bundle T ⊕T ∗ over Mfm 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]
One can show that Mfm-natural bilinear operators are regular because of the Peetre theorem
Summary
In the whole paper the word “bilinear” means “bilinear over R”.Let Mfm be the category of m-dimensional C∞ manifolds and their embeddings.The “doubled” tangent bundle T ⊕T ∗ over Mfm is full of interest because of the non-degenerate symmetric bilinear form and the Courant bracket, see [2]. In the present note, we deduce that if m ≥ p+1 ≥ 2 (or m = p ≥ 3, respectively), any Mfm-natural skew-symmetric bilinear operator A as above coincides with the generalized Courant bracket up to three (or two, respectively) real constants.
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