4 Vector bundles

Recall that we had the tangent bundle of a manifold. The tangent bundle gives

us a vector space at each point in space, namely the tangent space. In general,

a vector bundle is a vector space attached to each point in our manifold (in a

smoothly-varying way), which is what we are going to study in this chapter.

Before we start, we have a look at tensor products. These will provide us a

way of constructing new vector spaces from old ones.