Whitney embedding theorem
This article is about an embedding theorem, viz about sufficient conditions for a given manifold (with some additional structure) to be realized as an embedded submanifold of a standard space (real or complex projective or affine space)
View a complete list of embedding theorems
This fact is an application of the following pivotal fact/result/idea: Sard's theorem
View other applications of Sard's theorem OR Read a survey article on applying Sard's theorem
This fact is an application of the following pivotal fact/result/idea: existence of smooth partitions of unity
View other applications of existence of smooth partitions of unity OR Read a survey article on applying existence of smooth partitions of unity
The Whitney embedding theorem states the following:
- Any compact connected differential manifold of dimension , can be embedded inside
- Any compact connected differential manifold of dimension , can be immersed inside
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.