# Sheaf of sections of a vector bundle

From Diffgeom

## Definition

Let be a differential manifold, and be a smooth vector bundle over . The **sheaf of sections** of is defined as follows:

- To every open subset of , it associates the vector space of all smooth sections of the bundle
- 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 -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.