Nash Embedding Theorem

Nash's embedding theorem answers a fundamental question: can every abstract Riemannian manifold be realized as a submanifold of Euclidean space with the induced metric? The answer is yes, and the proof is one of the great achievements of 20th-century mathematics.

Statement

Theorem6.6Nash embedding theorem

Every compact Riemannian manifold $(M^n, g)$ admits a smooth isometric embedding into some Euclidean space $\mathbb{R}^N$ . Specifically:

(Nash 1956) If $M$ is compact, there exists a $C^k$ isometric embedding for $k \geq 3$ into $\mathbb{R}^N$ with $N = \frac{3n^3 + 14n^2 + 11n}{2}$ (later improved).
(Nash-Kuiper 1954-55, $C^1$ case) Any short embedding $f: (M,g) \to \mathbb{R}^{n+1}$ can be uniformly approximated by $C^1$ isometric embeddings. In particular, $S^2$ with the round metric can be $C^1$ -isometrically embedded into an arbitrarily small ball in $\mathbb{R}^3$ .

Context and Significance

RemarkIntrinsic vs. extrinsic geometry

Riemann originally defined metrics intrinsically (on abstract manifolds), while classical differential geometry studied surfaces embedded in $\mathbb{R}^3$ . Nash's theorem shows these viewpoints are equivalent: every intrinsic geometry can be extrinsically realized. However, the required ambient dimension $N$ may be much larger than $n$ .

Definition6.9Short embedding

An embedding $f: (M, g) \to (\mathbb{R}^N, g_{\mathrm{Eucl}})$ is short if $f^*g_{\mathrm{Eucl}} \leq g$ as bilinear forms, meaning lengths are not increased: $|df(v)| \leq |v|_g$ for all tangent vectors $v$ .

Proof Ideas

Proof

Nash's approach for the smooth case uses an implicit function theorem for nonlinear PDEs. The isometric embedding problem requires solving $f^*g_{\mathrm{Eucl}} = g$ , which in coordinates is $\sum_{a=1}^N \frac{\partial f^a}{\partial x^i} \frac{\partial f^a}{\partial x^j} = g_{ij}$ , a system of $\frac{n(n+1)}{2}$ equations in $N$ unknowns.

Key difficulty: The linearized problem loses derivatives (the linearization is not surjective in standard Sobolev or Holder spaces). Nash overcame this by inventing Nash-Moser iteration: a modified Newton's method that uses smoothing operators to compensate for the loss of derivatives at each step.

The $C^1$ case (Nash-Kuiper) uses a completely different approach: successive corrugation (wrinkling) to add small curvatures. The resulting embeddings are $C^1$ but highly non-smooth -- they are "crumpled" and exhibit fractal-like behavior. This is an early example of what is now called convex integration or $h$ -principle methods (Gromov). $\blacksquare$

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Impact

ExampleSurprising consequences

The $C^1$ Nash-Kuiper theorem produces counterintuitive results:

The unit sphere $S^2$ can be $C^1$ -isometrically embedded into a ball of radius $\epsilon$ in $\mathbb{R}^3$ . The embedding is continuous but not smooth -- it wrinkles infinitely.
A flat torus $T^2$ can be $C^1$ -isometrically embedded into $\mathbb{R}^3$ , despite the fact that no smooth isometric embedding exists (as shown by the Gauss-Bonnet theorem: a smooth surface in $\mathbb{R}^3$ with $K = 0$ everywhere must be a ruled surface, hence not compact).

RemarkNash-Moser theory

The Nash-Moser implicit function theorem, developed for the embedding problem, has become a fundamental tool in nonlinear analysis and PDE theory. It applies whenever the linearized problem exhibits a "loss of derivatives" phenomenon, including KAM theory in dynamical systems and the study of free boundary problems.