Nash embedding theorems

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The Nash embedding theorems (or imbedding theorems), named after

arclength
however the page is bent.

The first theorem is for

smooth
of class Ck, 3 ≤ k ≤ ∞. These two theorems are very different from each other. The first theorem has a very simple proof but leads to some counterintuitive conclusions, while the second theorem has a technical and counterintuitive proof but leads to a less surprising result.

The C1 theorem was published in 1954, the Ck-theorem in 1956. The real analytic theorem was first treated by Nash in 1966; his argument was simplified considerably by

contraction mapping theorem could be applied.[1]

Nash–Kuiper theorem (C1 embedding theorem)

Given an m-dimensional Riemannian manifold (M, g), an isometric embedding is a continuously differentiable

coordinate chart x) as a system of 1/2m(m + 1) many first-order partial differential equations
for n unknown (real-valued) functions:

If n is less than 1/2m(m + 1), then there are more equations than unknowns. From this perspective, the existence of isometric embeddings given by the following theorem is considered surprising.

Nash–Kuiper theorem.

short smooth embedding (or immersion) into Euclidean space ℝn, where n ≥ m + 1. This map is not required to be isometric. Then there is a sequence of continuously differentiable isometric embeddings (or immersions) M → ℝn of g which converge uniformly
to f.

The theorem was originally proved by John Nash with the stronger assumption n ≥ m + 2. His method was modified by Nicolaas Kuiper to obtain the theorem above.[3][4]

The isometric embeddings produced by the Nash–Kuiper theorem are often considered counterintuitive and pathological.

hyperbolic plane cannot be smoothly isometrically immersed into ℝ3. Any Einstein manifold of negative scalar curvature cannot be smoothly isometrically immersed as a hypersurface,[6] and a theorem of Shiing-Shen Chern and Kuiper even says that any closed m-dimensional manifold of nonpositive sectional curvature cannot be smoothly isometrically immersed in ℝ2m – 1.[7] Furthermore, some smooth isometric embeddings exhibit rigidity phenomena which are violated by the largely unrestricted choice of f in the Nash–Kuiper theorem. For example, the image of any smooth isometric hypersurface immersion of the round sphere must itself be a round sphere.[8] By contrast, the Nash–Kuiper theorem ensures the existence of continuously differentiable isometric hypersurface immersions of the round sphere which are arbitrarily close to (for instance) a topological embedding of the sphere as a small ellipsoid
.

Any closed and oriented two-dimensional manifold can be smoothly embedded in ℝ3. Any such embedding can be scaled by an arbitrarily small constant so as to become short, relative to any given Riemannian metric on the surface. It follows from the Nash–Kuiper theorem that there are continuously differentiable isometric embeddings of any such Riemannian surface where the radius of a circumscribed ball is arbitrarily small. By contrast, no negatively curved closed surface can even be smoothly isometrically embedded in ℝ3.[9] Moreover, for any smooth (or even C2) isometric embedding of an arbitrary closed Riemannian surface, there is a quantitative (positive) lower bound on the radius of a circumscribed ball in terms of the surface area and curvature of the embedded metric.[10]

In higher dimension, as follows from the Whitney embedding theorem, the Nash–Kuiper theorem shows that any closed m-dimensional Riemannian manifold admits an continuously differentiable isometric embedding into an arbitrarily small neighborhood in 2m-dimensional Euclidean space. Although Whitney's theorem also applies to noncompact manifolds, such embeddings cannot simply be scaled by a small constant so as to become short. Nash proved that every m-dimensional Riemannian manifold admits a continuously differentiable isometric embedding into ℝ2m + 1.[11]

At the time of Nash's work, his theorem was considered to be something of a mathematical curiosity. The result itself has not found major applications. However, Nash's method of proof was adapted by

h-principle.[13] This was applied by Stefan Müller and Vladimír Šverák to Hilbert's nineteenth problem, constructing minimizers of minimal differentiability in the calculus of variations.[14]

Ck embedding theorem

The technical statement appearing in Nash's original paper is as follows: if M is a given m-dimensional Riemannian manifold (analytic or of class Ck, 3 ≤ k ≤ ∞), then there exists a number n (with n ≤ m(3m+11)/2 if M is a compact manifold, and with n ≤ m(m+1)(3m+11)/2 if M is a non-compact manifold) and an

dot product
of Rn in the following sense:

for all vectors u, v in TpM. When n is larger than 1/2m(m + 1), this is an underdetermined system of partial differential equations (PDEs).

The Nash embedding theorem is a global theorem in the sense that the whole manifold is embedded into Rn. A local embedding theorem is much simpler and can be proved using the

Leonid Kantorovitch
.

Citations

  1. ^ Taylor 2011, pp. 147–151.
  2. ^ Eliashberg & Mishachev 2002, Chapter 21; Gromov 1986, Section 2.4.9.
  3. ^ Nash 1954.
  4. ^ Kuiper 1955a; Kuiper 1955b.
  5. ^ Kobayashi & Nomizu 1969, Note 18.
  6. ^ Kobayashi & Nomizu 1969, Theorem VII.5.3.
  7. ^ Kobayashi & Nomizu 1969, Corollary VII.4.8.
  8. ^ Kobayashi & Nomizu 1969, Corollary VII.5.4 and Note 15.
  9. ^ Kobayashi & Nomizu 1969, Theorem VII.5.6.
  10. ^ Burago & Zalgaller 1988, Corollary 6.2.2.
  11. ^ Nash 1954, pp. 394–395.
  12. ^ De Lellis & Székelyhidi 2013; Isett 2018.
  13. ^ Gromov 1986, Section 2.4.
  14. ^ Müller & Šverák 2003.
  15. ^ Nash 1956.

General and cited references

  • MR 0936419
    .
  • S2CID 2693636
    .
  • MR 1909245
    .
  • MR 0283728
    .
  • MR 0864505
    .
  • Günther, Matthias (1989). "Zum Einbettungssatz von J. Nash" [On the embedding theorem of J. Nash].
    MR 1037168
    .
  • Isett, Philip (2018). "A proof of Onsager's conjecture".
    S2CID 119267892. Archived from the original
    on 2022-10-11. Retrieved 2022-05-06.
  • MR 0238225
    .
  • MR 0075640
    .
  • MR 0075640
    .
  • S2CID 55855605
    .
  • MR 0065993
    .
  • MR 0075639. (Erratum:  [1]
    )
  • MR 0205266
    .
  • MR 2744149
    .
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