Primary Spectrum of C∞(M) and Jet Theory

We consider, for each smooth manifold M , the set M of all primary ideals of C (M ) which are closed and whose radical is maximal.The classical Lie theory of jets (jets of submanifolds) must be extended to M in order to have nice functorial properties.We will begin with the purely algebraic notions, referred always to the ring C (M ) .Subsequently, the differentiable structures on each jet space of a given type will be introduced.The theory of contact systems, which generalizes the classical one, has a purely algebraic part and another one which depends on the differentiable structures.