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 the following:

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

Related results

• Hard Whitney embedding theorem

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.