Sheaf of infinitely differentiable functions - Revision history

Vipul: 6 revisions

2008-05-18T20:09:15Z

6 revisions

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Vipul: /* Sheaf properties */

2007-12-26T21:33:41Z

Sheaf properties

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	* <math>f</math> does not have the same germ as the zero function, at the point <math>x</math>		* <math>f</math> does not have the same germ as the zero function, at the point <math>x</math>
	* For any open set <math>V</math> containing <math>x</math>, there exists a nonempty open set <math>W</math> contained in <math>V</math> such that <math>f</math> is identically zero on <math>W</math>		* For any open set <math>V</math> containing <math>x</math>, there exists a nonempty open set <math>W</math> contained in <math>V</math> such that <math>f</math> is identically zero on <math>W</math>

			The [[Hausdorffization]] of the sheaf looks locally like the [[sheaf of real-analytic functions]] (though there may be no global interpretation of this if the manifold does not have a real-analytic structure).

Vipul at 21:32, 26 December 2007

2007-12-26T21:32:23Z

← Older revision		Revision as of 21:32, 26 December 2007
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	* [[Sheaf of multiply differentiable functions]]		* [[Sheaf of multiply differentiable functions]]
			* [[Sheaf of real-analytic functions]]
			* [[Sheaf of continuous functions]]

			==Sheaf properties==

			{{soft sheaf}}

			The sheaf of infinitely differentiable functions on any [[differential manifold]] is a [[soft sheaf]]; in other words, given a differentiable function in an open neighbourhood of a closed subset, there is a differentiable function on the whole manifold whose germ on the closed subset is the same as that of the function on the open neighbourhood.

			{{non-Hausdorff etale}}

			The [[etale space]] associated to the sheaf of infinitely differentiable function on a manifold is not a [[Hausdorff space]]. This is a consequence of the fact that for any point <math>x \in M</math>, there exists an infinitely differentiable function <math>f</math> on a neighbourhood <math>U</math> of <math>x</math> such that:

			* <math>f</math> does not have the same germ as the zero function, at the point <math>x</math>
			* For any open set <math>V</math> containing <math>x</math>, there exists a nonempty open set <math>W</math> contained in <math>V</math> such that <math>f</math> is identically zero on <math>W</math>

Vipul at 20:29, 26 December 2007

2007-12-26T20:29:02Z

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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 infinitely 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)		* To every open set, we associate the ring of all [[infinitely differentiable function]]s (<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 infinitely 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.

			==Related notions==

			* [[Sheaf of multiply differentiable functions]]

Vipul: Sheaf of differentiable functions moved to Sheaf of infinitely differentiable functions: More accurate title

2007-12-26T20:27:34Z

Sheaf of differentiable functions moved to Sheaf of infinitely differentiable functions: More accurate title

← Older revision	Revision as of 20:27, 26 December 2007
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Vipul at 20:25, 26 December 2007

2007-12-26T20:25:28Z

← Older revision		Revision as of 20:25, 26 December 2007
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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.

Vipul at 09:28, 27 August 2007

2007-08-27T09:28:36Z

New page

==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 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.