Manifolds of differentiable maps

Abstract

1. Let X, Y be smooth finite dimensional manifolds, let C^∞(X,Y) be the set of smooth mappings from X to Y; for any non negative integer n let Jⁿ(X,Y) denote the fibre bundle of n-jets of smooth maps from X to Y, equipped with the canonical manifold structure which makes jⁿf : X → Jⁿ(X,Y) into a smooth section for each f ∈c^∞(X,Y) , where jⁿf(x) is the n-jet of f at xϵ X.

Usually C^∞(X,Y) is equipped with the so called Whitney-C^∞-topology: a basis of open sets is given by all sets of the form M(U) = {f ϵC^∞(X,Y) : jⁿf (X)⊆ U} , where U is any open set in Jⁿ(X,Y) and nϵN. See [ 3] and [6] for accounts of this topology. We may describe it intuitively by the following words: if you go to infinity on X you may control better and better partial derivatives up to a fixed order.