Sheaf of multiply differentiable functions

From Diffgeom
Jump to: navigation, search
This article describes a sheaf on a manifold (the manifold may possess some additional structure in terms of which the sheaf is defined)
View other sheaves on manifolds

Definition

Let M be a C^r-manifold. The sheaf of C^r-functions' on M, also called the sheaf of r times differentiable functions, is defined as follows:

  • To each open set U, it associates the ring (or rather \R-algebra) of C^r-functions on U
  • The restriction is the usual function restriction

The sheaf of C^r-functions in fact completely encodes the C^r-structure. In other words, given a topological manifold with a sheaf of functions that is supposed to be the sheaf of C^r-functions, the C^r-structure on the manifold is dictated by the sheaf.

Facts

Étale space is non-Hausdorff

The étale space of the shead of multiply differentiable functions is not Hausdorff. This is essentially because there can be C^r-functions which look the same in one direction but different in others.

The Hausdorffization of the étale space yields the space of polynomial functions of degree at most r.

Further information: Hausdorffization of étale space of multiply differentiable functions