# 20. Sections of Vector Bundles

A fibre bundle $ \pi \colon E \to M$ is a surjective submersion between manifolds with the property that the domain $E$ has extra structure. Smooth maps that go in the opposite direction are—from the point of view of fibre bundles—uninteresting unless they respect this extra structure. Those that do are called *sections*.

In the case of a vector bundle, the space of sections is a module over the ring of smooth functions. The main result of today’s lecture is the so-called Hom-Gamma Theorem^{[1]} that relates vector bundle homomorphisms and operators on the space of sections that are linear over the smooth functions:

$$ \Gamma \big( \mathrm{Hom}(E, F) \big) \cong \mathrm{Hom}\big( \Gamma(E) ,\Gamma(F)\big).$$

Along the way we introduce the notion of a *point operator* and a *local operator* between spaces of sections.

The lecture concludes with another algebraic interlude, which covers the rudiments of *sheaf theory*. This material is *for interest only*, and can safely be skipped — it will not be used in the remainder of the course.

Roughly speaking, a *presheaf* is a way to assign data locally to open subsets of a topological space in such a way that it is compatible with restrictions. A *sheaf* is a presheaf for which it is possible to go backwards and reassemble global data from local data.

After giving the relevant definitions, we give a “sheaf-theoretic” definition of a manifold and a vector bundle, and outline the proof that there is an equivalence of categories between the category of vector bundles over $M$ and the category of finite rank locally free $C^{ \infty}(M)$-modules.

Where by “so-called” what I actually mean is “called by me”. I made the name up ¯\_(ツ)_/¯ ↩︎