Nash embedding theorems: Difference between revisions

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{{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 '''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 three-dimensional Euclidean space because curves drawn on the page retain the same [[arclengtharc length]] however the page is bent.

The first theorem is for [[continuously differentiable]] (''C''¹) embeddings and the second for embeddings that are [[analytic function|analytic]] or [[smooth function|smooth]] of class ''C^k'', 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 first theorem is for [[continuously differentiable]] ({{math|''C''¹}}) embeddings and the second for embeddings that are [[analytic function|analytic]] or [[smooth function|smooth]] of class {{math|''C''^''k''''}}, {{math|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''¹ theorem was published in 1954, the ''C^k''-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^k''- 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.<ref>{{cite book|first=Michael E.|last=Taylor|author-link=Michael E. Taylor|title=Partial Differential Equations III: Nonlinear equations|mr=2744149 |edition = 2nd |series=Applied Mathematical Sciences|volume= 117|publisher= Springer|year= 2011|isbn=978-1-4419-7048-0|chapter=Isometric imbedding of Riemannian manifolds|pages=147–151}}</ref>

The ''C''¹ theorem was published in 1954, and the {{math|''C''^''k''''-}} 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 {{math|''C''^''k''''-}} 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.<ref>{{cite booksfnm|first=Michael E.|last1a1=Taylor|author-link1y=Michael E. Taylor|title=Partial Differential Equations III: Nonlinear equations|mr=2744149 |edition = 2nd |series=Applied Mathematical Sciences|volume= 117|publisher= Springer|year= 2011|isbn=978-1-4419-7048-0|chapter=Isometric imbedding of Riemannian manifolds|pages1pp=147–151}}</ref>

==Nash–Kuiper theorem (''C''¹ embedding theorem) {{anchor|Nash–Kuiper theorem}}==

'''Theorem.''' Let (''M'',''g'') be a Riemannian manifold and ƒ: ''M^m'' → '''R'''^''n'' a [[short map|short]] ''C''^∞-embedding (or [[Immersion (mathematics)|immersion]]) into Euclidean space '''R'''^''n'', where ''n'' ≥ ''m''+1. Then for arbitrary ε > 0 there is an embedding (or immersion) ƒ_ε: ''M^m'' → '''R'''^''n'' which is

== Nash–Kuiper theorem ({{math|''C''¹}} embedding theorem) {{<span class="anchor|" id="Nash–Kuiper theorem}}"></span> ==

{{ordered list|type=lower-roman

Given an {{mvar|m}}-dimensional Riemannian manifold {{math|(''M'', ''g'')}}, an ''isometric embedding'' is a continuously differentiable [[topological embedding]] {{math|''f'' : ''M'' → '''R'''^''n''}} 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:

| in class ''C''¹,

:<math>g_{ij}(x)=\sum_{\alpha=1}^n\frac{\partial f^\alpha}{\partial x^i}\frac{\partial f^\alpha}{\partial x^j}.</math>

| isometric: for any two vectors ''v'',''w''&nbsp;&isin;&nbsp;''T_x''(''M'') in the [[tangent space]] at ''x'' &isin; ''M'',

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.

:<math>g(v,w)=\langle df_\epsilon(v),df_\epsilon(w)\rangle</math>,

<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'''^''n''}} a [[short map|short]] smooth embedding (or [[Immersion (mathematics)|immersion]]) into Euclidean space {{math|'''R'''^''n''}}, 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'''^''n''}} of {{mvar|g}} which [[uniform convergence|converge uniformly]] to {{mvar|f}}.</blockquote>

| &epsilon;-close to ƒ:

:<math> | f(x) - f_\epsilon (x) | < \epsilon ~\forall~ x\in M</math>.

}}

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}}

In particular, as follows from the [[Whitney embedding theorem]], any ''m''-dimensional Riemannian manifold admits an isometric ''C''¹-embedding into an ''arbitrarily small neighborhood'' in 2''m''-dimensional Euclidean space.

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'''³}}. 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'''^2''m''–1}}.{{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]].

The theorem was originally proved by John Nash with the condition ''n'' ≥ ''m''+2 instead of ''n'' ≥ ''m''+1 and generalized by [[Nicolaas Kuiper]], by a relatively easy trick.

Any closed and oriented two-dimensional manifold can be smoothly embedded in {{math|'''R'''³}}. 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'''³}}.{{sfnm|1a1=Kobayashi|1a2=Nomizu|1y=1969|1loc=Theorem VII.5.6}} Moreover, for any smooth (or even {{math|''C''²}}) 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}}

The theorem has many counterintuitive implications. For example, it follows that any closed oriented Riemannian surface can be ''C''¹ isometrically embedded into an arbitrarily small [[ball (mathematics)|&epsilon;-ball]] in Euclidean 3-space (for small <math>\epsilon</math> there is no such ''C''²-embedding since from the [[Gaussian curvature#Alternative formulas|formula for the Gauss curvature]] an extremal point of such an embedding would have curvature ≥ &epsilon;⁻²). And, there exist ''C''¹ isometric embeddings of the hyperbolic plane in '''R'''³.

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'''^2''m''+1}}.{{sfnm|1a1=Nash|1y=1954|1pp=394–395}}

==''C''^''k'' 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^k'', 3 ≤ ''k'' ≤ ∞), then there exists a number ''n'' (with ''n'' ≤ ''m''(3''m''+11)/2, if ''M'' is a compact manifold ''n'' ≤ ''m''(''m''+1), or (3''m''+11)/2 if ''M'' is a non-compact manifold) and an [[isometric embedding]] ƒ: ''M'' → '''R'''^''n'' (also analytic or of class ''C^k'').<ref>{{Cite journal|last=Nash|first=John|date=January 1956|title=The Imbedding Problem for Riemannian Manifolds|url=https://www.jstor.org/stable/1969989|journal=The Annals of Mathematics|volume=63|issue=1|pages=20|doi=10.2307/1969989|jstor=1969989}}</ref> That is ƒ is an [[Embedding#Differential topology|embedding]] of ''C^k'' manifolds and for every point ''p'' of ''M'', the [[derivative]] dƒ_''p'' is a [[linear operator|linear map]] from the [[tangent space]] ''T_pM'' to '''R'''^''n'' which is compatible with the given [[inner product space|inner product]] on ''T_pM'' and the standard [[scalar product|dot product]] of '''R'''^''n'' in the following sense:

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''^''k'' embedding theorem ==

The technical statement appearing in Nash's original paper is as follows: if {{math|''M''}} is a given {{math|''m''}}-dimensional Riemannian manifold (analytic or of class {{math|''C''^''k''''}}, {{math|3 ≤ ''k'' ≤ ∞}}), then there exists a number ''n'' (with {{math|''n'' ≤ ''m''(3''m'' + 11)/2,}} if ''M'' is a compact manifold, and with {{nowrap|''n'' ≤ ''m''(''m'' + 1), or (3''m'' + 11)/2}} if ''M'' is a non-compact manifold) and an [[isometric embedding]] ƒ{{math|''f'' : ''M'' → '''R'''^''n''}} (also analytic or of class ''C^k'').<ref>{{Cite journalsfnm|last1a1=Nash|first1y=John|date=January 1956|title=The Imbedding Problem for Riemannian Manifolds|url=https://www.jstor.org/stable/1969989|journal=The Annals of Mathematics|volume=63|issue=1|pages=20|doi=10.2307/1969989|jstor=1969989}}</ref> That is ƒ{{math|''f''}} is an [[Embedding#Differential topology|embedding]] of {{math|''C''^''k''''}} manifolds and for every point {{math|''p''}} of {{math|''M''}}, the [[derivative]] dƒ{{math|d''f''_''p''}} is a [[linear operator|linear map]] from the [[tangent space]] {{math|''T''_''p''''M''}} to {{math|'''R'''^''n''}} whichthat is compatible with the given [[inner product space|inner product]] on {{math|''T''_''p''''M''}} and the standard [[scalar product|dot product]] of {{math|'''R'''^''n''}} in the following sense:

: <math>\langle u,v \rangle = df_p(u)\cdot df_p(v)</math>

for all vectors {{math|''u''}}, {{math|''v''}} in {{math|''T''_''p''''M''}}. ThisWhen {{mvar|n}} is larger than {{math|{{sfrac|1|2}}''m''(''m'' + 1)}}, this is an undeterminedunderdetermined system of [[partial differential equation]]s (PDEs).<ref>In a [https://web.math.princeton.edu/jfnj/texts_and_graphics/Main.Content/Erratum.txt 1998 email correspondence] with the logician [[Robert M. Solovay]], Nash mentioned an error in his original argument for bounding the dimension of the embedding space for the case of non-compact manifolds.</ref>

The Nash embedding theorem is a global theorem in the sense that the whole manifold is embedded into {{math|'''R'''^''n''}}. 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 far-reachingimplicit generalizationfunction theorem for isometric embeddings. This theorem has been generalized by a number of theother implicitauthors functionto theoremabstract contexts, thewhere it is known as [[Nash–Moser theorem]] and Newton's method with postconditioning. The basic idea in the proof of Nash's solutionimplicit offunction the embedding problemtheorem is the use of [[Newton's method]] to prove the existence of a solution to the above system ofconstruct PDEssolutions. 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 Thereother iscontexts, also an older method calledthe [[Kantorovich theorem|Kantorovichconvergence iteration]]of thatthe usesstandard Newton's method]] directlyhad (withoutearlier thebeen introductionproved ofby smoothing[[Leonid operators)Kantorovitch]].

== NotesSee also ==

* {{annotated link|Representation theorem}}

* {{annotated link|Whitney embedding theorem}}

* {{annotated link|Whitney immersion theorem}}

* {{annotated link|Takens's theorem}}

* {{annotated link|Nonlinear dimensionality reduction}}

* {{annotated link|Universal space}}

== Citations ==

{{reflist|30em}}

== References ==

== General and cited references ==

* {{citation|last1=Greene|first1=Robert E.|author1-link= Robert Everist Greene |last2 = Jacobowitz|first2=Howard|title= Analytic Isometric Embeddings|journal=[[Annals of Mathematics]]|volume=93|pages=189–204|doi=10.2307/1970760|issue=1|year=1971|jstor=1970760|mr=0283728}}

{{refbegin}}

* {{citation|first=Matthias|last=Günther|title=Zum Einbettungssatz von J. Nash | trans-title=On the embedding theorem of J. Nash | language=German |

* {{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}}

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}}

* {{citationcite journal |firstlast1=NicolaasDe HendrikLellis|lastfirst1=KuiperCamillo|authorlinklast2=NicolaasSzékelyhidi|first2=László KuiperJr.|title=OnDissipative ''C''¹-isometriccontinuous imbeddings.Euler Iflows|journal=Indagationes[[Inventiones Mathematicae (Proceedings)]]|volume=58193|year=19552013|issue=2|pages=545–556377–407|mr=00756403090182|author-link1=Camillo De Lellis|doi=10.10161007/S1385s00222-7258(55)50075012-80429-9| 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}}

* {{citation|first=Nicolaas Hendrik|last=Kuiper|authorlink=Nicolaas Kuiper|title=On ''C''¹-isometric imbeddings. II|journal=Indagationes Mathematicae (Proceedings)|volume=58|year=1955|pages=683–689|mr=0075640|doi=10.1016/S1385-7258(55)50093-X}}

* {{citationcite journal |firstlast1=JohnGreene|lastfirst1=NashRobert E.|authorlinkauthor1-link=John ForbesRobert Nash,Everist Greene Jr.|last2 = Jacobowitz|first2=Howard|title=''C''¹- Analytic isometric imbeddingsembeddings|journal=[[Annals of Mathematics]]|volumeseries=60Second Series|yearvolume=195493|pages=383–396189–204|doi=10.2307/19698401970760|issue=31|year=1971|jstor=19698401970760|mr=00659930283728}}.

* {{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}}

* {{citation|first=John|last=Nash|authorlink=John Forbes Nash, Jr.|title=The imbedding problem for Riemannian manifolds|journal=[[Annals of Mathematics]]|volume=63|year=1956|pages=20–63|doi=10.2307/1969989|issue=1|mr=0075639|jstor=1969989}}.

* {{citationcite journal |first=Matthias|last=Günther|title=Zum Einbettungssatz von J. Nash | issue=1|trans-title=On the embedding theorem of J. Nash | language=German |

* {{citation|first=John|last=Nash|title=Analyticity of the solutions of implicit function problem with analytic data|authorlink=John Forbes Nash, Jr.|journal=[[Annals of Mathematics]]|volume=84|year=1966|pages=345–355|doi=10.2307/1970448|issue=3|jstor=1970448|mr=0205266}}.

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|url-access=subscription}}

* {{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''¹}}-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}}

* {{citationcite journal|first=Nicolaas HendrikH.|last=Kuiper|authorlink=Nicolaas Kuiper|title=On {{math|''C''¹}}-isometric imbeddings. II|journal=[[Indagationes Mathematicae|Indagationes Mathematicae (Proceedings)]]|volume=58|year=19551955b|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''¹}} 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}}

* {{citationwikicite|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}}}}

* {{citationcite journal |first=JohnJ.|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}}