Sheaf of smooth sections

Let $\mathbb F\in\{\mathbb R,\mathbb C\}$ , and let $\pi:E\to M$ be a smooth finite-rank $\mathbb F$ -vector bundle. Its sheaf of smooth sections, denoted $\mathcal E$ or $\Gamma^\infty(-,E)$ , assigns

\mathcal E(U)=\Gamma^\infty(U,E|_U)

to each open subset $U\subseteq M$ . Restriction of sections supplies the restriction maps. Sections that agree on overlaps glue uniquely, and smoothness is local, so this is a sheaf.

Pointwise scalar multiplication makes $\mathcal E$ a sheaf of modules over the sheaf $C^\infty_M(\mathbb F)$ , defined by

C^\infty_M(\mathbb F)(U)=C^\infty(U,\mathbb F).

For $\mathbb F=\mathbb R$ , this is $C^\infty_M$ ; for $\mathbb F=\mathbb C$ , it is its complexification.

Local freeness

If $E|_U\cong U\times\mathbb F^r$ , a local frame identifies

\mathcal E|_U\cong \bigl(C^\infty_M(\mathbb F)|_U\bigr)^{\oplus r}.

Hence the section sheaf is a finite-rank locally free sheaf. Its stalk $\mathcal E_x$ consists of germs of local sections near $x$ ; it is a free module of rank $r$ over the local ring $C^\infty_{M,x}(\mathbb F)$ , the stalk of $C^\infty_M(\mathbb F)$ at $x$ .

Sheaf versus global module

The module of global smooth sections is the single module

\mathcal E(M)=\Gamma^\infty(M,E)

over $C^\infty(M,\mathbb F)$ . The sheaf $\mathcal E$ includes sections over every open set and their gluing data. Local freeness of $\mathcal E$ does not assert that $\Gamma^\infty(M,E)$ is a free module.

A bundle morphism $E\to F$ covering $\operatorname{id}_M$ induces a morphism $\mathcal E\to\mathcal F$ of $C^\infty_M$ -module sheaves by postcomposition.

References

John M. Lee, Introduction to Smooth Manifolds, 2nd ed., Springer, 2012. DOI record. Relevant: vector bundles, local frames, and smooth sections.
Dale Husemoller, Fibre Bundles, 3rd ed., Springer, 1994. DOI record. Relevant: vector bundles and their local trivializations.