Introduction

The mathematical motivation for studying vector bundles comes from the example of the tangent bundle $TM$ of a manifold $M$. Recall that the tangent bundle is the union of all the tangent spaces $T_{m}M$ for every $m$ in $M$. As such it is a collection of vector spaces, one for every point of $M$.

The physical motivation comes from the realisation that the fields in physics may not just be maps $\phi: M \to \mathbb{C}^N$ say, but may take values in different vector spaces at each point. Tensors do this for example. The argument for this is partly quantum mechanics because, if $\phi$ is a wave function on a space-time $M$ say, then what we can know about are expectation values, that is things like:

$$\int_M \langle \phi(x) , \phi(x) \rangle dx$$

and to define these all we need to know is that $\phi(x)$ takes its values in a one-dimensional complex vector space with Hermitian inner product. There is no reason for this to be the same one-dimensional Hermitian vector space here as on Alpha Centauri. Functions like $\phi$, which are generalisations of complex valued functions, are called sections of vector bundles. We will consider first the simplest theory of vector bundles where the vector space is a one-dimensional complex vector space - line bundles.