# Sheaf of derivations of a manifold

From Diffgeom

This article defines a sheaf that can be associated to a differential manifold. The global analog of this sheaf, which is also the same as the object of the sheaf associated to the whole manifold, is:Lie algebra of global derivations

## Definition

### Definition in terms of the tangent bundle

Let be a differential manifold. The **sheaf of derivations** of is defined as the sheaf of smooth sections of the tangent bundle of the manifold. In other words:

- For every open subset of , the associated object is the vector space of all smooth sections of the tangent bundle on , i.e. smooth vector fields on
- The restriction map is the restriction of a vector field from a larger open subset to a smaller open subset

### Definition in terms of algebraic theory of derivations

Let be a differential manifold. The **sheaf of derivations** of is defined as the algebra-theoretic sheaf of derivations for the sheaf of infinitely differentiable functions on .