On the natural operators on vector fields (Q1124179)

Local product preserving functors from the category of manifolds into itself are given by the action of Weil-algebras A, by \(T_ A(M)=Hom(C^{\infty}(M),A)\). In this paper all natural (with respect to local diffeomorphisms of the base manifold M) lifts of vector fields on M to vector fields on \(T_ A(M)\) are determined. The answer is as follows. There is the flow prolongation: differentiate the local 1-parameter group functor applied to the flow. On this an arbitrary element of the Weil- algebra may act as generalized scalar multiplication. To this one may add one of the generalized Liouville vector fields, which are natural: they correspond exactly to the automorphisms of the Weil algebra.