Differential geometry Riemannian and pseudo-Riemannian manifolds Theorem

Nash embedding theorems

Builds on Riemannian manifold Riemannian metric

Statement

Let $(M, g)$ be a Riemannian manifold of dimension $m$ . Write $\ip{\cdot}{\cdot}$ for the Euclidean inner product on $\R^n$ . An isometric embedding of $(M, g)$ into $\R^n$ is an embedding $f \colon M \to \R^n$ with $\ip{df_p(v)}{df_p(w)} = g_p(v, w)$ for all $p \in M$ and $v, w \in T_pM$ .

( $C^1$ theorem, Nash and Kuiper.) Let $n \ge m + 1$ and let $f \colon M \to \R^n$ be a smooth embedding that is short, that is $\abs{df_p(v)}^2 < g_p(v, v)$ for all $p$ and all nonzero $v \in T_pM$ . Then for every $\varepsilon > 0$ there is a $C^1$ embedding $\tilde f \colon M \to \R^n$ that is isometric for $g$ and satisfies $\abs{\tilde f(p) - f(p)} < \varepsilon$ for all $p \in M$ .
( $C^k$ theorem, Nash.) Let $3 \le k \le \infty$ and let $g$ be of class $C^k$ . Then $(M, g)$ has an isometric embedding of class $C^k$ into $\R^n$ . Here $n = m(3m + 11)/2$ if $M$ is compact, and $n = m(m + 1)(3m + 11)/2$ otherwise.

Remarks

Part 2 is proved by solving the nonlinear system $f^*\ip{\cdot}{\cdot} = g$ with an iteration that combines Newton's method with smoothing operators, the Nash–Moser technique. Part 1 is proved by convex integration: one adds many small fast oscillations that increase lengths in a controlled way. An example of part 1: the flat square torus $\R^2 / \Z^2$ has a $C^1$ isometric embedding into $\R^3$ . It has no $C^2$ isometric embedding there, because a compact surface in $\R^3$ has a point of positive Gaussian curvature, while the flat torus has curvature $0$ . So the low codimension in part 1 really needs the low regularity $C^1$ . Nash proved part 1 with $n \ge m + 2$ , and Kuiper extended it to $n \ge m + 1$ .

Going deeper

For a student

The Whitney embedding theorem embeds a manifold in $\R^{2m}$ but ignores the metric; Nash's theorems make the embedding respect the metric at the cost of more dimensions or less smoothness. The $C^1$ theorem is an early instance of the h-principle: a short embedding can be deformed into an isometric one by arbitrarily small moves. The bounds on $n$ in part 2 have been lowered by later work, but the existence statement is what is used in practice.

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Tag	`8YQJQXQHVE`
URI	`m2://8YQJQXQHVE`
Kind	statement, theorem
Writing	W0 stub
MSC	53C42, 53B20
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About	Riemannian manifold
Uses	Riemannian manifold, Riemannian metric
Same as	Wikidata Q1306092 Wikipedia Nash embedding theorems