In this chapter, we describe the basic structure upon which our study of jets will be based, namely that of bundles and sections. This structure is a generalisation of the more familiar structure of pairs of manifolds and maps, and allows more complicated topological arrangements. Although we shall be concerned primarily with local properties of jets, this more general description is still necessary for our discussion, because there are pairs of manifolds whose jet bundles do not themselves simplify to further pairs of manifolds.

Fibred Manifolds and Bundles

Many of the theories in modern mathematical physics can be described by considering smooth functions between differentiable manifolds. The domain of such a function might represent a region of space-time, and the codomain the possible states of the relevant physical system. Frequently, however, one considers not the function itself, but rather its graph: if the function is f : M → F then its graph is the new function grf : M → M × F defined by grf(p) = (p, f(p)), and any function φ : M → M × F which satisfies the condition pr1 ◦ φ = idM is the graph of a uniquely-defined function f (namely, f = pr2 ◦ φ). In this arrangement, the product manifold M × F is called the total space, because its local coordinate charts contain both dependent and independent variables for the function f. The domain M is also called the base space.