Whitney embedding theorem - Revision history

Vipul: 7 revisions

2008-05-18T20:13:31Z

7 revisions

← Older revision	Revision as of 20:13, 18 May 2008
(No difference)

Vipul at 21:37, 2 April 2008

2008-04-02T21:37:25Z

← Older revision		Revision as of 21:37, 2 April 2008
Line 1:		Line 1:
	{{embedding theorem~~\|[[differential manifold]]s~~}}		{{embedding theorem}}

	{{applicationof\|Sard's theorem}}		{{applicationof\|Sard's theorem}}

Vipul at 21:36, 2 April 2008

2008-04-02T21:36:34Z

← Older revision		Revision as of 21:36, 2 April 2008
Line 12:		Line 12:
	* Any compact connected differential manifold of dimension <math>n</math>, can be [[immersion\|immersed]] inside <math>\R^{2n}</math>		* Any compact connected differential manifold of dimension <math>n</math>, can be [[immersion\|immersed]] inside <math>\R^{2n}</math>

			==Related results==

			* [[Hard Whitney embedding theorem]]
	==Proof==		==Proof==

Vipul at 21:28, 2 April 2008

2008-04-02T21:28:10Z

← Older revision		Revision as of 21:28, 2 April 2008
Line 7:		Line 7:
	==Statement==		==Statement==

	The '''Whitney embedding theorem''' states ~~that any~~ compact connected differential manifold of dimension <math>n</math> ~~possesses a smooth embedding into~~ <math>\R^{2n+1}</math>~~. By smooth embedding~~, ~~we mean it~~ can be ~~viewed as a subspace, with the subspace topology, and further, that the induced mapping of tangent spaces is also injective.~~		The '''Whitney embedding theorem''' states the following:

			* Any compact connected differential manifold of dimension <math>n</math>, can be embedded inside <math>\R^{2n +1}</math>
			* Any compact connected differential manifold of dimension <math>n</math>, can be [[immersion\|immersed]] inside <math>\R^{2n}</math>

	==Proof==		==Proof==

Vipul at 21:26, 2 April 2008

2008-04-02T21:26:50Z

← Older revision		Revision as of 21:26, 2 April 2008
Line 3:		Line 3:
	{{applicationof\|Sard's theorem}}		{{applicationof\|Sard's theorem}}

	{{~~applicationof~~\|existence of smooth partitions of unity}}		{{application of\|existence of smooth partitions of unity}}

	==Statement==		==Statement==

Vipul at 23:53, 16 January 2008

2008-01-16T23:53:40Z

← Older revision		Revision as of 23:53, 16 January 2008
Line 1:		Line 1:
	{{embedding theorem\|[[differential manifold]]s}}		{{embedding theorem\|[[differential manifold]]s}}

			{{applicationof\|Sard's theorem}}

			{{applicationof\|existence of smooth partitions of unity}}

	==Statement==		==Statement==

129.199.2.17 at 05:52, 23 May 2007

2007-05-23T05:52:39Z

← Older revision		Revision as of 05:52, 23 May 2007
Line 1:		Line 1:
	{{embedding theorem}}		{{embedding theorem\|[[differential manifold]]s}}

	==Statement==		==Statement==

129.199.2.17 at 05:51, 23 May 2007

2007-05-23T05:51:54Z

New page

{{embedding theorem}}

==Statement==

The '''Whitney embedding theorem''' states that any compact connected differential manifold of dimension <math>n</math> possesses a smooth embedding into <math>\R^{2n+1}</math>. By smooth embedding, we mean it can be viewed as a subspace, with the subspace topology, and further, that the induced mapping of tangent spaces is also injective.

==Proof==

===Proof ingredients===

Two ingredients are used in the proof:

* Compactness allows us to work with a finite atlas, and consider a [[partition of unity]]

* [[Sard's theorem]], or rather, the following corollary of Sard's theorem: if <math>m < n</math>, the image of any <math>m</math>-dimensional manifold in a <math>n</math>-dimensional manifold via a differentiable map, has measure zero in the latter.
We can use Sard's theorem to predict certain properties of maps that we construct.