Nash theorems (in differential geometry)

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