Whitney embedding theorem: Difference between revisions
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Two ingredients are used in the proof: | Two ingredients are used in the proof: | ||
* Compactness | * 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. | ||
Revision as of 05:46, 23 May 2007
Statement
The Whitney embedding theorem states that any compact connected differential manifold of dimension possesses a smooth embedding into . 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 , the image of any -dimensional manifold in a -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.