Sheaf of infinitely differentiable functions: Difference between revisions
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{{sheaf on manifold}} | |||
==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) | * 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 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.