In Chapter 11 we described how the notion of parallelism in an affine space or on a surface may be extended to apply to any differentiable manifold to give a theory of parallel translation of vectors, which in general is path-dependent. An associated idea is that of covariant differentiation, which generalises the directional derivative operator in an affine space, considered as an operator on vector fields. We used the word “connection” to stand for this collection of ideas.
In Chapter 13 we showed that a connection on a manifold has an alternative description in terms of a structure on its tangent bundle, namely, a distribution of horizontal subspaces, a curve in the tangent bundle having everywhere horizontal tangent vector if it represents a curve in the base with a parallel vector field along it.
In Chapter 14 we defined vector bundles. These spaces share some important properties with tangent bundles (which are themselves examples of vector bundles), namely linearity of the fibre, and the existence of local bases of sections. It is natural to ask whether the idea of a connection may be extended to vector bundles in general, so as to define notions of parallelism and of directional differentiation of (local) sections of a vector bundle. We shall show in this chapter how this may be done, first by adapting the rules of covariant differentiation on a manifold, and then, at a deeper level, by defining a structure not on the vector bundle itself but rather on a principal bundle with which it is associated.