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

Statement

The Whitney embedding theorem states that any compact connected differential manifold of dimension $n$ possesses a smooth embedding into ${\mathbb{R}}^{2n+1}$. 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 $m<n$, 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.