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
.