The Contact System for A-jet manifolds

Abstract. Jets of a manifold M can be described as ideals of C1(M). This way, all the usual processes on jets can be directly referred to that ring. By using this fact, we give a very simple construction of the contact system on jet spaces. The same way, we also dene the contact system for the recently considered A-jet spaces, where A is a Weil algebra. We will need to introduce the concept of derived algebra. Although without formalization, jets are present in the work of S. Lie (see, for instance, [6]; x 130, pp. 541) who does not assume a bered structure on the concerned manifold; on the contrary, this assumption is usually done nowadays in the more narrow approach given by the jets of sections. It is an old idea to consider the points of a manifold other than the ordinary ones. This can be traced back to Plucker, Grassmann, Lie or Weil. Jets are `points' of a manifoldM and can be described as ideals of its ring of dierentiable functions [9, 13]. Indeed, the k-jets of m-dimensional submanifolds of M are those ideals p C1(M) such that C1(M)=p is isomorphic to Rk