Whitney embedding theorem

Remarks

There is also a weak version: every such $M$ embeds in $\R^{2n+1}$, and any smooth map $M \to \R^{2n+1}$ can be approximated by embeddings. Its proof starts from an embedding into some large $\R^N$, built from charts and bump functions. Then $N$ is lowered by generic linear projections. Sard's theorem shows that almost every projection stays an injective immersion while $N > 2n + 1$. To reach $2n$, one first takes an immersion into $\R^{2n}$ with only transverse double points. Then one removes the double points in pairs; for $n \ge 3$ this step uses the Whitney trick, and $n = 1, 2$ are treated directly. The bound $2n$ is sharp in general: for $n$ a power of $2$, the real projective space $\R P^n$ does not embed in $\R^{2n - 1}$. Example: the circle ($n = 1$) embeds in $\R^2$, and the Klein bottle ($n = 2$) embeds in $\R^4$ but not in $\R^3$.

Going deeper

For a student

The theorem reconciles the intrinsic definition of a manifold by charts with the older picture of a manifold as a subset of $\R^N$. It allows one to borrow structures from $\R^N$, such as a Riemannian metric induced from the dot product or a normal bundle and a tubular neighbourhood. The weak version is a typical general position argument, a precursor of transversality theory. The strong version introduced the Whitney trick, which later became the key step in the h-cobordism theorem.