• In this chapter we begin the part of the book in which the concept of the fiber bundle enters the story. Bundles play a significant role in modern geometry and their language as well as techniques are widely used in modern theoretical and mathematical physics. That is why the strategy of ignoring them, although in principle possible, would be fairly short-sighted. In the forthcoming two chapters we will look in some detail at two particular bundles closely associated with Lagrangian and Hamiltonian mechanics, the tangent and cotangent bundle. These (as well as numerous further) bundles may be canonically constructed for an arbitrary manifold M and one can find a fairly rich geometry on them, resulting “free of charge” directly from the way they are defined. Moreover, this additional geometrical structure turns out to be just what is needed for the formulation of the two versions (Lagrangian and Hamiltonian) of classical mechanics. Later on (in Chapter 19) another bundle will be introduced, which may be canonically assigned to an arbitrary manifold M, the frame bundle. It provides a novel view of a linear connection on M. Generalizing the three bundles we will then introduce the concepts of the principal G-bundle and the associated bundle, which turn out to be the essential ingredients needed for the development of the theory of connections and gauge fields.

From a didactic point of view it is convenient to begin to study some particular bundle and to notice its relevant features, which then enter the official abstract definition of the concept of a bundle. The two particular bundles which suit our purpose well are the tangent and cotangent bundles.