View source for Sheaf of infinitely differentiable functions

{{sheaf on manifold}}

==Definition==

Let <math>M</math> be a [[differential manifold]]. The '''sheaf of differentiable functions''' of <math>M</math> is defined as follows:

* To every open set, we associate the ring of all differentiable functions (<math>C^\infty</math>-functions) from that open set to the real numbers (the ring structure arises from pointwise operations)
* The restriction map is simply function restriction

In fact, a differential manifold is completely characterized by its sheaf of differentiable functions. In other words, given a [[topological manifold]] and the sheaf of differentiable functions arising from some choice of differential structure on it, the differential manifold structure can be recovered from the sheaf.