Sheaf of derivations of a manifold

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

• For every open subset $U$ of $M$, the associated object is the vector space of all smooth sections of the tangent bundle on $U$, i.e. smooth vector fields on $U$
• 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 $M$ be a differential manifold. The sheaf of derivations of $M$ is defined as the algebra-theoretic sheaf of derivations for the sheaf of infinitely differentiable functions on $M$.