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:

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