Liftings of 1-forms to \((J^r T^*)^*\) (Q2773277)

Let \(F\) be a natural bundle over \(n\)-manifolds; a linear natural differential operator \(A\colon T^*\to T^*F\) (``lifting of \(1\)-forms to \(F\)'') is a system of \(\mathbb R\)-linear maps \(A\colon \Omega^1(M)\to \Omega^1(FM)\) for every \(n\)-dimensional manifold \(M\), which is compatible with the embeddings of \(n\)-manifolds [see \textit{I. Kolar, P. Michor} and \textit{J. Slovak}, `Natural operations in differential geometry' (Springer-Verlag, Berlin) (1993; Zbl 0782.53013)].NEWLINENEWLINENEWLINEIn this article a complete classification of these operators is given in the case of \(F\) being \(FM:=(J^rT^*M)^*\) (the dual of the \(r\)-jet bundle of the cotangent bundle).NEWLINENEWLINENEWLINEThe computation is based on a previous result by the author: If \(n\geq 2\) (resp. \(n=1\)), any linear natural differential operator \(A\) is determined by \(A(x^2dx^1)\) (resp. \(A(dx^1)\)) where \(x^1,x^2\) belong to a system of coordinates on \({\mathbb R}^n\) [\textit{W. M. Mikulski}, Arch. Math., Brno 31, 97-111 (1995; Zbl 0844.58006)].NEWLINENEWLINENEWLINEOn the other hand, there are three basic examples of such liftings: \(A^V\), which consists of pull-back of \(1\)-forms from \(M\) to \(FM\); \(A^{[r]}\), which is related to the \(r\)-jet prolongation of forms and, \(A^{(1)}\), which is associated with a certain first-order natural linear differential operator \(T^*\to J^1T^*\).NEWLINENEWLINENEWLINEWhen \(n\geq 2\), one introduces the local diffeomorphism \(G\) defined in some open neighborhood of \(0\in{\mathbb R}^n\) by \(G^*x^1=x^1-(x^2)^2/2\), \(G^*x^2=x^2/(1-x^1)\), \(G^*x^i=x^i\), \(i\geq 3\); an analysis of the action of \(G\) on \(A\) and the aforementioned result are applied by the author in order to state the main theorem which describes all the possible liftings of \(1\)-forms as follows (the case \(n=1\) is studied separately):NEWLINENEWLINENEWLINEIf \(r\geq 2\) and \(n\geq 2\) or if \(n=1\) (resp. if \(r=1\) and \(n\geq 2\)) then the vector space of linear natural differential operators \(A\colon T^*\to T^*(J^rT^*)^*\) is spanned by \(A^{[r]}\) and \(A^V\) (resp. by \(A^{[1]}\), \(A^{(1)}\) and \(A^V\)).NEWLINENEWLINENEWLINEAs an application, the classification of all linear natural transformations \(J^rT^*\to J^rT^*\) is obtained.