On the Green--Vinogradov formula
An explicit and global way to construct a primitive of the exact form appearing in the general Green formula (Green-Vinogradov formula) of Vinogradov [\textit{A. M. Vinogradov}, Acta Appl. Math. 2, 21-78 (1984; Zbl 0534.58005)] for a given linear connection \(\nabla\), is given. Let \(M\) be a smooth manifold, \(A=C^\infty(M)\), \(P,Q\) \(A\)-modules of sections of vector bundles on \(M\), and \(\Delta:P\to Q\) a differential operator. Let \(\widehat P\) be \(P^*\otimes \Lambda^n\), \(\langle\cdot, \cdot\rangle_P\) the pairing \(P\otimes \widehat P\to\Lambda^n\) and \(\Delta_{p,\widehat q}=\widehat q\circ\Delta\circ p\). Then the adjoint operator \(\Delta^*\) of \(\Delta\) is defined by \(\langle \Delta^* (\widehat q),p\rangle_P =(\Delta_{p,\widehat q})^*(1) \in\Lambda^n\) (Def. 3.5). The Green-Vinogradov formula takes the form \[ \bigl\langle \Delta (p),\widehat q\bigr\rangle =\bigl\langle p,\Delta^*(\widehat q)\bigr \rangle+ d\psi_\nabla (\Delta_{p,\widehat q}). \] In this paper, by using Spencer sequence (explained in section 4), the decomposition morphism is introduced (Def. 5.1) and it is shown each linear connection \(\nabla\) determines a decomposition morphism \(\lambda_\nabla\) (Th. 5.4). Then it is shown \[ \psi_\nabla (\Delta_{p, \widehat q})= \lambda_\nabla \bigl(\nu (\Delta_{p,\widehat q})\bigr)^*,\tag{1} \] where \(\nu (\Delta)= \Delta-\Delta (1)\) (Th. 6.2). Local computations of \(\psi_\nabla (\Delta_{p,\widehat q})\) for general 4-th order operators and so on are also given. \(\psi_\nabla (\Delta_{p,\widehat q})\) is a conserved current of \(\Delta =0\). In this respect, the paper is concluded by giving a brief outline of the application of the Green-Vinogradov formula to finding conservation laws (Th. 6.5).
scientific article; zbMATH DE number 1783451