Vector Bundles

Abstract

In earlier chapters, we defined tangent spaces at individual points of a manifold and described vector fields X as collections of tangent vectors X p, p ∈ M depending smoothly on the base point. Similarly, 1-forms α were seen as collections of covectors α p, p ∈ M depending smoothly on the base point. We will now explain that the tangent bundle given as the disjoint union of all tangent spaces and similarly the cotangent bundle given as the disjoint union of all cotangent spaces are manifolds in their own right. This then allows us to interpret vector fields and 1-forms as smooth maps from the base manifold into these bundles. In fact, both are special cases of vector bundles—collections of vector spaces labeled by points of the base manifold, with smooth dependence on the base points. All the natural constructions and concepts of linear algebra—such as direct sums, duals, quotient spaces, tensor products, inner product spaces, and so on—extend to vector bundles in a natural way, compatible with the smooth structures.