Whitney embedding theorem - Revision history

Vipul: 2 revisions

2008-05-11T20:00:20Z

2 revisions

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Vipul: /* Proof ingredients */

2007-05-23T05:46:27Z

Proof ingredients

← Older revision		Revision as of 05:46, 23 May 2007
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	Two ingredients are used in the proof:		Two ingredients are used in the proof:

	* Compactness ~~helps~~ us ~~transform the original problem~~ to ~~one~~ with a finite atlas		* 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.		* [[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.		We can use Sard's theorem to predict certain properties of maps that we construct.

Vipul at 05:40, 23 May 2007

2007-05-23T05:40:52Z

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 helps us transform the original problem to one with a finite atlas

* [[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.