# Nash theorems (in differential geometry)

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Two groups of theorems on isometrically imbedded and immersed Riemannian manifolds in a Euclidean space (see also Immersion of a manifold; Isometric immersion). The original versions are due to J. Nash ([1]).

1. Nash’s theorem on $C^{1}$-imbeddings and $C^{1}$-immersions. A $C^{1}$-immersion (-imbedding) $f: V^{n} \to E^{m}$ of an $n$-dimensional Riemannian space $V^{n}$ with a $C^{0}$-metric $g$ into an $m$-dimensional Euclidean space $E^{m}$ is called short if and only if the metric $g_{f}$ induced by it on $V^{n}$ is such that the quadratic form $g - g_{f}$ is positive definite. If $V^{n}$ has a short immersion (imbedding) into $E^{m}$, where $m \geq n + 1$, then $V^{n}$ also has an isometric $C^{1}$-immersion (-imbedding) into $E^{m}$. Under the restriction $m \geq n + 2$, this theorem was proved in [1], and in the form stated above in [2]. This theorem implies, in particular, that if a compact Riemannian manifold $V^{n}$ has a $C^{1}$-imbedding (-immersion) into $E^{m}$, where $m \geq n + 1$, then $V^{n}$ also has an isometric $C^{1}$-imbedding (-immersion) into $E^{m}$. Another consequence of Nash’s theorem is that every point of $V^{n}$ has a sufficiently small neighborhood that admits an isometric $C^{1}$-imbedding into $E^{n + 1}$.
2. Nash’s theorem on regular imbeddings. Every compact Riemannian manifold $V^{n}$ of class $C^{r}$, where $3 \leq r \leq \infty$, has an isometric $C^{r}$-imbedding into $E^{m}$, where $m = \dfrac{3 n^{2} + 11 n}{2}$. If $V^{n}$ is not compact, then it has an isometric $C^{r}$-imbedding into $E^{m_{1}}$, where $m_{1} = \dfrac{(3 n^{2} + 11) (n + 1)}{2}$.

Nash’s theorem on regular imbeddings results from an application of Nash’s implicit-function theorem on the inversion of a broad class of differential operators. The meaning of this theorem is that when a certain linear algebraic system of equations connected naturally with a differential operator $L$ is solvable, and when a reasonable topology is introduced in the image and inverse image, then the operator in question is an open mapping, i.e., $L$ is locally invertible near any point of its range. For the equations of an imbedding of a Riemannian manifold into a Euclidean space, this reduces to the fact that the first and second derivatives of the mapping $f: V^{n} \to E^{m}$ with respect to the intrinsic coordinates of $V^{n}$ must be linearly independent. Such imbeddings were first considered in [4]; they are called free. Nash’s implicit-function theorem implies that a compact Riemannian manifold $V^{n}$ that is sufficiently close to another one $W^{n}$ having a free imbedding into $E^{m}$ also has a free imbedding into $E^{m}$. This fact, and the original method of extension with respect to a parameter, lead to Nash’s theorem on regular imbeddings (see [3]). By extending Nash’s method to non-compact manifolds and analytic imbeddings, and also by a principal refinement of the process of extension with respect to a parameter, it has been proved that every infinitely differentiable (analytic) Riemannian manifold $V^{n}$ has an isometric differentiable (analytic) imbedding into $E^{m}$, where $m = \dfrac{n (n + 1)}{2} + 3 n + 5$ (see [5][7]).

#### References

 [1] J. Nash, “$C^{1}$-isometric imbeddings”, Ann. of Math., 60 (1954), pp. 383–396. MR0065993 Zbl 0058.37703 [2] N. Kuiper, “On $C^{1}$-isometric imbeddings”, Proc. K. Ned. Akad. Wetensch., A58: 4 (1955), pp. 545–556. MR75640 [3] J. Nash, “The imbedding problem for Riemannian manifolds”, Ann. of Math., 63 (1956), pp. 20–63. MR0075639 Zbl 0070.38603 [4] C. Burstin, “Ein Beitrag zum Problem der Einbettung der Riemannschen Räume in euklidischen Räumen”, Mat. Sb., 38: 3–4 (1931), pp. 74–85. Zbl 0006.08004 Zbl 57.0549.01 [5] J. Nash, “Analyticity of the solutions of implicit function problems with analytic data”, Ann. of Math., 84 (1966), pp. 345–355. MR0205266 Zbl 0173.09202 [6] M.L. Gromov, V.A. Rokhlin, “Embeddings and immersions in Riemannian geometry”, Russian Math. Surveys, 25: 5 (1970), pp. 1–57; Uspekhi Mat. Nauk, 25 (1970), pp. 53–62. Zbl 0222.53053 Zbl 0202.21004 [7] M.L. Gromov, “Isometric imbeddings and immersions”, Soviet Math. Dokl., 11: 3 (1970), pp. 1206–1209; Dokl. Akad. Nauk SSSR, 192 (1970), pp. 794–797. MR0275456 Zbl 0214.50404

#### Comments

The Nash theorem in differential topology says that a compact connected $C^{\infty}$-manifold without boundary is diffeomorphic to a component of a real algebraic variety.

Let $\pi: X \to V$ be a smooth (i.e., $C^{\infty}$-) fibration. Denote by ${J^{r}}(V,X)$ the space of $r$-jets (of germs) of smooth sections $f: V \to X$. The $r$-th order jet of a section $f: V \to X$ is denoted by $J^{r} f: V \to {J^{r}}(V,X)$. A section $\phi: V \to {J^{r}}(V,X)$ is called holonomic if and only if there is a $C^{r}$-section $f: V \to X$ such that $\phi = J^{r} f$; note that $\phi$ determines $f$ uniquely (if it exists). The fine topology on the space ${C^{0}}(V,X)$ of $C^{0}$-sections $f: V \to X$ is obtained by taking as a basis the subsets ${C^{0}}(V,U)$, where $U$ runs over the open subsets of $X$. The fine $C^{r}$-topology on ${C^{r}}(V,X)$ is induced by the imbedding $\left\{ \begin{matrix} {C^{r}}(V,X) & \to & {C^{0}}(V,{J^{r}}(V,X)) \\ f & \mapsto & J^{r} f \end{matrix} \right\}$ from the fine $C^{0}$-topology to ${C^{0}}(V,{J^{r}}(V,X))$.

The Nash approximation theorem says that an arbitrary Riemannian $C^{r}$-metric $g$ on $V$ has a fine $C^{r}$-approximation by some $C^{r}$-metric $g'$ on $V$ that admits $C^{r}$-immersions $f': (V,g') \to \mathbf{R}^{2 l}$ for some $l = l(n) < \infty$, where $n = \dim(V)$.

The Nash–Kuiper theorem ([1], [2]) says that an arbitrary differentiable immersion $f_{0}: V \to \mathbf{R}^{q}$ for $q > \dim(V)$ admits a $C^{1}$-continuous homotopy $(f_{t}: V \to \mathbf{R}^{q})_{t \in [0,1]}$ of immersions to an isometric immersion $f_{1}: V \to \mathbf{R}^{q}$.

#### References

 [a1] M.W. Hirsch, “Differential topology”, Springer (1976). MR0448362 Zbl 0356.57001 [a2] M. Gromov, “Partial differential relations”, Springer (1986). (Translated from Russian) MR0864505 Zbl 0651.53001
How to Cite This Entry:
Nash theorems (in differential geometry). Encyclopedia of Mathematics. URL: http://www.encyclopediaofmath.org/index.php?title=Nash_theorems_(in_differential_geometry)&oldid=40187
This article was adapted from an original article by D.D. Sokolov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article