View source for Nash embedding theorems

{{Short description|Every Riemannian manifold can be isometrically embedded into some Euclidean space}}

The '''Nash embedding theorems''' (or '''imbedding theorems'''), named after [[John Forbes Nash Jr.]], state that every [[Riemannian manifold]] can be isometrically [[embedding|embedded]] into some [[Euclidean space]]. [[Isometry|Isometric]] means preserving the length of every [[rectifiable path|path]].  For instance, bending but neither stretching nor tearing a page of paper gives an [[isometric embedding]] of the page into Euclidean space because curves drawn on the page retain the same [[arclength]] however the page is bent.

The first theorem is for [[continuously differentiable]] (''C''<sup>1</sup>) embeddings and the second for embeddings that are [[analytic function|analytic]] or [[smooth function|smooth]] of class ''C<sup>k</sup>'', 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 ''C''<sup>1</sup> theorem was published in 1954, the ''C<sup>k</sup>''-theorem in 1956.  The real analytic theorem was first treated by Nash in 1966; his argument was simplified considerably by {{harvtxt|Greene|Jacobowitz|1971}}. (A local version of this result was proved by [[Élie Cartan]] and [[Maurice Janet]] in the 1920s.)  In the real analytic case, the smoothing operators (see below) in the Nash inverse function argument can be replaced by Cauchy estimates.  Nash's proof of the ''C<sup>k</sup>''- case was later extrapolated into the [[h-principle]] and [[Nash–Moser theorem|Nash–Moser implicit function theorem]].  A simpler proof of the second Nash embedding theorem was obtained by {{harvtxt|Günther|1989}} who reduced the set of nonlinear [[partial differential equation]]s to an elliptic system, to which the [[contraction mapping theorem]] could be applied.{{sfnm|1a1=Taylor|1y=2011|1pp=147–151}}

{{Anchor|Nash–Kuiper theorem}}
==Nash–Kuiper theorem ({{math|''C''<sup>1</sup>}} embedding theorem) ==
Given an {{mvar|m}}-dimensional Riemannian manifold {{math|(''M'', ''g'')}}, an ''isometric embedding'' is a continuously differentiable [[topological embedding]] {{math|''f'': ''M'' → ℝ<sup>''n''</sup>}} such that the [[pullback]] of the Euclidean metric equals {{mvar|g}}. In analytical terms, this may be viewed (relative to a smooth [[coordinate chart]] {{mvar|x}}) as a system of {{math|{{sfrac|1|2}}''m''(''m'' + 1)}} many first-order [[partial differential equation]]s for {{mvar|n}} unknown (real-valued) functions:
:<math>g_{ij}(x)=\sum_{\alpha=1}^n\frac{\partial f^\alpha}{\partial x^i}\frac{\partial f^\alpha}{\partial x^j}.</math>
If {{mvar|n}} is less than {{math|{{sfrac|1|2}}''m''(''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.
<blockquote>'''Nash–Kuiper theorem.'''{{sfnm|1a1=Eliashberg|1a2=Mishachev|1y=2002|1loc=Chapter 21|2a1=Gromov|2y=1986|2loc=Section 2.4.9}} Let {{math|(''M'', ''g'')}} be an {{mvar|m}}-dimensional Riemannian manifold and {{math|''f'': ''M'' → ℝ<sup>''n''</sup>}} a [[short map|short]] smooth embedding (or [[Immersion (mathematics)|immersion]]) into Euclidean space {{math|ℝ<sup>''n''</sup>}}, where {{math|''n'' ≥ ''m'' + 1}}. This map is not required to be isometric. Then there is a sequence of continuously differentiable isometric embeddings (or immersions) {{math|''M'' → ℝ<sup>''n''</sup>}} of {{mvar|g}} which [[uniform convergence|converge uniformly]] to {{mvar|f}}.</blockquote>

The theorem was originally proved by John Nash with the stronger assumption {{math|''n'' ≥ ''m'' + 2}}. His method was modified by [[Nicolaas Kuiper]] to obtain the theorem above.{{sfnm|1a1=Nash|1y=1954}}{{sfnm|1a1=Kuiper|1y=1955a|2a1=Kuiper|2y=1955b}}

The isometric embeddings produced by the Nash–Kuiper theorem are often considered counterintuitive and pathological.{{sfnm|1a1=Kobayashi|1a2=Nomizu|1y=1969|1loc=Note 18}} They often fail to be smoothly differentiable. For example, a [[Hilbert's theorem (differential geometry)|well-known theorem]] of [[David Hilbert]] asserts that the [[hyperbolic plane]] cannot be smoothly isometrically immersed into {{math|ℝ<sup>3</sup>}}. Any [[Einstein manifold]] of negative [[scalar curvature]] cannot be smoothly isometrically immersed as a hypersurface,{{sfnm|1a1=Kobayashi|1a2=Nomizu|1y=1969|1loc=Theorem VII.5.3}} and a theorem of [[Shiing-Shen Chern]] and Kuiper even says that any [[closed manifold|closed]] {{mvar|m}}-dimensional manifold of nonpositive [[sectional curvature]] cannot be smoothly isometrically immersed in {{math|ℝ<sup>2''m'' – 1</sup>}}.{{sfnm|1a1=Kobayashi|1a2=Nomizu|1y=1969|1loc=Corollary VII.4.8}} Furthermore, some smooth isometric embeddings exhibit rigidity phenomena which are violated by the largely unrestricted choice of {{mvar|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.{{sfnm|1a1=Kobayashi|1a2=Nomizu|1y=1969|1loc=Corollary VII.5.4 and Note 15}} 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 {{math|ℝ<sup>3</sup>}}. 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 {{math|ℝ<sup>3</sup>}}.{{sfnm|1a1=Kobayashi|1a2=Nomizu|1y=1969|1loc=Theorem VII.5.6}} Moreover, for any smooth (or even {{math|''C''<sup>2</sup>}}) 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.{{sfnm|1a1=Burago|1a2=Zalgaller|1y=1988|1loc=Corollary 6.2.2}}

In higher dimension, as follows from the [[Whitney embedding theorem]], the Nash–Kuiper theorem shows that any closed {{mvar|m}}-dimensional Riemannian manifold admits an continuously differentiable isometric embedding into an ''arbitrarily small neighborhood'' in {{math|2''m''}}-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 {{mvar|m}}-dimensional Riemannian manifold admits a continuously differentiable isometric embedding into {{math|ℝ<sup>2''m'' + 1</sup>}}.{{sfnm|1a1=Nash|1y=1954|1pp=394–395}}

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 [[Camillo De Lellis]] and László Székelyhidi to construct low-regularity solutions, with prescribed [[kinetic energy]], of the [[Euler equation]]s from the mathematical study of [[fluid mechanics]]. In analytical terms, the Euler equations have a formal similarity to the isometric embedding equations, via the quadratic nonlinearity in the first derivatives of the unknown function.{{sfnm|1a1=De Lellis|1a2=Székelyhidi|1y=2013|2a1=Isett|2y=2018}} The ideas of Nash's proof were abstracted by [[Mikhael Gromov (mathematician)|Mikhael Gromov]] to the principle of ''convex integration'', with a corresponding [[h-principle]].{{sfnm|1a1=Gromov|1y=1986|1loc=Section 2.4}} This was applied by [[Stefan Müller (mathematician)|Stefan Müller]] and [[Vladimír Šverák]] to [[Hilbert's nineteenth problem]], constructing minimizers of minimal differentiability in the [[calculus of variations]].{{sfnm|1a1=Müller|1a2=Šverák|1y=2003}}

==''C''<sup>''k''</sup> 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 ''C<sup>k</sup>'', 3 ≤ ''k'' ≤ ∞), then there exists a number ''n'' (with ''n'' ≤ ''m''(3''m''+11)/2 if ''M'' is a compact manifold, and with ''n'' ≤ ''m''(''m''+1)(3''m''+11)/2 if ''M'' is a non-compact manifold) and an [[isometric embedding]] ƒ: ''M'' → '''R'''<sup>''n''</sup> (also analytic or of class ''C<sup>k</sup>'').{{sfnm|1a1=Nash|1y=1956}} That is ƒ is an [[Embedding#Differential topology|embedding]] of ''C<sup>k</sup>'' manifolds and for every point ''p'' of ''M'', the [[derivative]] dƒ<sub>''p''</sub> is a [[linear operator|linear map]] from the [[tangent space]] ''T<sub>p</sub>M'' to '''R'''<sup>''n''</sup> which is compatible with the given [[inner product space|inner product]] on ''T<sub>p</sub>M'' and the standard [[scalar product|dot product]] of '''R'''<sup>''n''</sup> in the following sense:
: <math>\langle u,v \rangle = df_p(u)\cdot df_p(v)</math>
for all vectors ''u'', ''v'' in ''T<sub>p</sub>M''. When {{mvar|n}} is larger than {{math|{{sfrac|1|2}}''m''(''m'' + 1)}}, this is an underdetermined system of [[partial differential equation]]s (PDEs).

The Nash embedding theorem is a global theorem in the sense that the whole manifold is embedded into '''R'''<sup>''n''</sup>.  A local embedding theorem is much simpler and can be proved using the [[implicit function theorem]] of advanced calculus in a [[Manifold#Charts|coordinate neighborhood]] of the manifold. The proof of the global embedding theorem relies on Nash's implicit function theorem for isometric embeddings. This theorem has been generalized by a number of other authors to abstract contexts, where it is known as [[Nash–Moser theorem]]. The basic idea in the proof of Nash's implicit function theorem is the use of [[Newton's method]] to construct solutions. The standard Newton's method fails to converge when applied to the system; Nash uses smoothing operators defined by [[convolution]] to make the Newton iteration converge: this is Newton's method with postconditioning.  The fact that this technique furnishes a solution is in itself an [[existence theorem]] and of independent interest. In other contexts, the [[Kantorovich theorem|convergence of the standard Newton's method]] had earlier been proved by [[Leonid Kantorovitch]].

== Citations ==
{{Reflist|30em}}

== General and cited references ==
{{refbegin}}
* {{cite book|last1=Burago|first1=Yu. D.|last2=Zalgaller|first2=V. A.|title=Geometric inequalities|others=Translated from the Russian by A. B. Sosinskiĭ|series=Grundlehren der mathematischen Wissenschaften|volume=285|publisher=[[Springer-Verlag]]|location=Berlin|year=1988|isbn=3-540-13615-0|mr=0936419|author-link1=Yuri Burago|author-link2=Victor Zalgaller|doi=10.1007/978-3-662-07441-1}}
* {{cite journal|last1=De Lellis|first1=Camillo|last2=Székelyhidi|first2=László Jr.|title=Dissipative continuous Euler flows|journal=[[Inventiones Mathematicae]]|volume=193|year=2013|issue=2|pages=377–407|mr=3090182|author-link1=Camillo De Lellis|doi=10.1007/s00222-012-0429-9| arxiv=1202.1751 | bibcode=2013InMat.193..377D | s2cid=2693636 }}
* {{cite book|last1=Eliashberg|first1=Y.|last2=Mishachev|first2=N.|title=Introduction to the h-principle|series=[[Graduate Studies in Mathematics]]|volume=48|publisher=[[American Mathematical Society]]|location=Providence, RI|year=2002|isbn=0-8218-3227-1|mr=1909245|author-link1=Yakov Eliashberg|doi=10.1090/gsm/048}}
* {{cite journal|last1=Greene|first1=Robert E.|author1-link= Robert Everist Greene |last2 = Jacobowitz|first2=Howard|title= Analytic isometric embeddings|journal=[[Annals of Mathematics]]|series=Second Series|volume=93|pages=189–204|doi=10.2307/1970760|issue=1|year=1971|jstor=1970760|mr=0283728}}
* {{cite book|last1=Gromov|first1=Mikhael|title=Partial differential relations|series=Ergebnisse der Mathematik und ihrer Grenzgebiete (3)|volume=9|publisher=[[Springer-Verlag]]|location=Berlin|year=1986|isbn=3-540-12177-3|mr=0864505|author-link1=Mikhael Gromov (mathematician)|doi=10.1007/978-3-662-02267-2}}
* {{cite journal|first=Matthias|last=Günther|title=Zum Einbettungssatz von J. Nash | issue=1|trans-title=On the embedding theorem of J. Nash | language=German |
journal=[[Mathematische Nachrichten]]|volume= 144 |year=1989|pages= 165–187|doi=10.1002/mana.19891440113 | mr=1037168|url = https://onlinelibrary.wiley.com/doi/abs/10.1002/mana.19891440113}}
* {{cite journal|last1=Isett|first1=Philip|title=A proof of Onsager's conjecture|journal=[[Annals of Mathematics]]|series=Second Series|year=2018|volume=188|issue=3|pages=871–963|mr=3866888|doi=10.4007/annals.2018.188.3.4|s2cid=119267892|url=https://authors.library.caltech.edu/87369/|arxiv=1608.08301|access-date=2022-05-06|archive-date=2022-10-11|archive-url=https://web.archive.org/web/20221011050610/https://authors.library.caltech.edu/87369/|url-status=dead}}
* {{cite book|mr=0238225|last1=Kobayashi|first1=Shoshichi|last2=Nomizu|first2=Katsumi|author-link2=Katsumi Nomizu|title=Foundations of differential geometry. Vol II|series=Interscience Tracts in Pure and Applied Mathematics|volume=15|title-link=Foundations of differential geometry|publisher=[[John Wiley & Sons, Inc.]]|location=New York–London|year=1969|others=Reprinted in 1996| issue=2 |isbn=0-471-15732-5|author-link1=Shoshichi Kobayashi}}
* {{cite journal|first=Nicolaas H.|last=Kuiper|authorlink=Nicolaas Kuiper|title=On {{math|''C''<sup>1</sup>}}-isometric imbeddings. I|journal=[[Indagationes Mathematicae|Indagationes Mathematicae (Proceedings)]]|volume=58|year=1955a|pages=545–556|mr=0075640|doi=10.1016/S1385-7258(55)50075-8}}
* {{cite journal|first=Nicolaas H.|last=Kuiper|authorlink=Nicolaas Kuiper|title=On {{math|''C''<sup>1</sup>}}-isometric imbeddings. II|journal=[[Indagationes Mathematicae|Indagationes Mathematicae (Proceedings)]]|volume=58|year=1955b|pages=683–689|mr=0075640|doi=10.1016/S1385-7258(55)50093-X}}
*{{cite journal|last1=Müller|first1=S.|last2=Šverák|first2=V.|title=Convex integration for Lipschitz mappings and counterexamples to regularity|journal=[[Annals of Mathematics]]|series=Second Series|volume=157|year=2003|issue=3|pages=715–742|mr=1983780|author-link1=Stefan Müller (mathematician)|author-link2=Vladimir Šverák|doi=10.4007/annals.2003.157.715| s2cid=55855605 |doi-access=free|arxiv=math/0402287}}
* {{cite journal|first=John|last=Nash|authorlink=John Forbes Nash, Jr.|title={{math|''C''<sup>1</sup>}} isometric imbeddings|journal=[[Annals of Mathematics]]|series=Second Series|volume=60|year=1954|pages=383–396|doi=10.2307/1969840|issue=3|jstor=1969840|mr=0065993}}
* {{wikicite|ref={{sfnRef|Nash|1956}}|reference={{cite journal|first=John|last=Nash|authorlink=John Forbes Nash, Jr.|title=The imbedding problem for Riemannian manifolds|journal=[[Annals of Mathematics]]|series=Second Series|volume=63|year=1956|pages=20–63|doi=10.2307/1969989|issue=1|mr=0075639|jstor=1969989|ref=none}} {{erratum|https://web.math.princeton.edu/jfnj/texts_and_graphics/Main.Content/Erratum.txt|checked=yes}}}}
* {{cite journal|first=J.|last=Nash|title=Analyticity of the solutions of implicit function problem with analytic data|authorlink=John Forbes Nash, Jr.|journal=[[Annals of Mathematics]]|series=Second Series|volume=84|year=1966|pages=345–355|doi=10.2307/1970448|issue=3|jstor=1970448|mr=0205266}}
* {{cite book|first=Michael E.|last=Taylor|author-link=Michael E. Taylor|title=Partial differential equations III. Nonlinear equations|mr=2744149 |edition = Second edition of 1996 original|series=Applied Mathematical Sciences|volume= 117|publisher= [[Springer Publishing|Springer]]|location=New York|year= 2011|isbn=978-1-4419-7048-0|doi=10.1007/978-1-4419-7049-7}}
{{refend}}

{{Riemannian geometry}}
{{Manifolds}}

[[Category:Riemannian geometry]]
[[Category:Riemannian manifolds]]
[[Category:Theorems in Riemannian geometry]]