The only global contact transformations of order two or more are point transformations

摘要

Let us consider J(m)(k)M as the space of k-jets of m-dimensional submanifolds of a smooth manifold M. Our purpose is to show that every contact transformation of J(m)(k)M, k greater than or equal to 2, is induced by a diffeomorphism of M (point transformations). It is also derived that a first order contact transformation can not be globally prolonged to higher order jets except when it is a point transformation. This holds true as well for jets of sections of a regular projection. The Legendre transformation gives us an example of this property.