Whitney embedding theorem

From Diffgeom
Jump to: navigation, search
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

Statement

The Whitney embedding theorem states the following:

  • Any compact connected differential manifold of dimension n, can be embedded inside \R^{2n +1}
  • Any compact connected differential manifold of dimension n, can be immersed inside \R^{2n}

Related results

Proof

Proof ingredients

Two ingredients are used in the proof:

  • Sard's theorem, or rather, the following corollary of Sard's theorem: if m < n, the image of any m-dimensional manifold in a n-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.