Difference between revisions of "Sheaf of infinitely differentiable functions"

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{{sheaf on manifold}}
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==Definition==
 
==Definition==
  
 
Let <math>M</math> be a [[differential manifold]]. The '''sheaf of differentiable functions''' of <math>M</math> is defined as follows:
 
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 from that open set to the real numbers (the ring structure arises from pointwise operations)
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* 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
 
* 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.
 
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.

Revision as of 20:25, 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 M be a differential manifold. The sheaf of differentiable functions of M is defined as follows:

  • To every open set, we associate the ring of all differentiable functions (C^\infty-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.