# Difference between revisions of "Sheaf of infinitely differentiable functions"

From Diffgeom

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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) | + | * 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.