7 - Infinite Jet Bundles

Summary

Many of the constructions described in the last chapter may be carried out on jet manifolds of various orders, with results which are related by the jet projections. In many cases, a clearer formulation of these results is possible if we can avoid the need to keep track of the order of the jets. The way to do this is to use “infinite jets”.

There are two approaches to this idea. One is to regard the “infinite jet manifold” as merely a convenient fiction, and to regard entities defined on different jet manifolds as equivalent when they are related by the appropriate projection maps; these equivalence classes are then the corresponding entities defined on the fictitious manifold “J^∞π”. With this approach, one has to keep in mind just which properties the various entities are meant to possess: for example, a “vector field” on “J^∞π” is actually an equivalence class of vector fields, and there is no reason a priori why such an object should have any of the standard properties of vector fields.

The alternative approach, which we shall adopt here, is to define J^∞π as a bona fide manifold. The result, of course, will be an infinite-dimensional manifold, and in the first section of this chapter we shall describe some of the ideas which are needed for its definition.

Preliminaries

The first two definitions in this section are taken from the theory of categories, although we shall only apply that theory to the particular category of real topological vector spaces and continuous linear maps.