Difference between revisions of "Sheaf of infinitely differentiable functions"
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Revision as of 20:27, 26 December 2007
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 be a differential manifold. The sheaf of differentiable functions of is defined as follows:
 To every open set, we associate the ring of all differentiable functions (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.