Sheaf of sections of a vector bundle

Definition

Let $M$ be a differential manifold, and $\pi :E\to M$ be a smooth vector bundle over $M$. The sheaf of sections of $E$ is defined as follows:

• To every open subset $U$ of $M$, it associates the vector space of all smooth sections of the bundle ${\pi }^{-1}(U)\to U$
• The restriction map is section restriction

Facts

• The sheaf of sections for the trivial one-dimensional bundle over a differential manifold is the sheaf of infinitely differentiable functions. This is more than just a sheaf of vector spaces: it is in fact a sheaf of $\mathbb{R}$-algebras.
• The sheaf of sections for the tangent bundle is the sheaf of derivations (derivations are also termed vector fields). This is the same as the algebra-theoretic sheaf of derivations of the sheaf of infinitely differentiable functions.
• The sheaf of sections for the cotangent bundle is the sheaf of differential 1-forms.