# 16. 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 “$ \mathrm{Hom-}\Gamma$ Theorem” that relates vector bundle homomorphisms and operators on the space of sections that are linear over the smooth functions:

$$ \Gamma ( \mathrm{Hom}(E_1, E_2)) \cong \mathrm{Hom}( \Gamma(E_1) ,\Gamma(E_2)).$$

Comments and questions?