6 - Vector Bundles

Summary

The notion of vector bundle gives a powerful and flexible tool for all kinds of calculus on differential manifolds. Important special cases are the tangent bundle of a differential manifold, the cotangent bundle, exterior powers of these bundles, etc. This chapter consists mostly of definitions and constructions.

Local Vector Bundles

Suppose M is a differential manifold of dimension n, for example, an open subset of Rⁿ, and V is an arbitrary k-dimensional vector space. The product manifold M × V, together with the “projection map” π: M × V → M such that π(p, v) = p, is called a local vector bundle of rankk over M. We call M the base manifold, and {p} × V is called the fiber overp, for any p ∈ M; think of a fiber as a copy of the vector space V, sitting on top of the point p, as in the picture on the next page. A convenient general notation is to refer to such a vector bundle as E = M × V, and to refer to the fiber over p as E_p = {p} × V.

Sections of Local Vector Bundles

A C^r (resp., C^∞) section of a local vector bundle E over M means a C^r mapping σ: M → M × R^k such that σ(p) ∈ E_p for every p. The idea is simple: σ chooses a point σ(p) in the fiber over p for every p, in a way that varies r-times differentiably across the fibers.