Nash embedding theorems: Difference between revisions

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

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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 theorem was originally proved by John Nash with the condition {{math|''n'' ≥ ''m'' + 2}}. His method was modified by [[Nicolaas Kuiper]] to allow {{math|''n'' {{=}} ''m'' + 1}}.{{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|ℝ³}}. 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 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.

Any closed and oriented two-dimensional manifold can be smoothly embedded in {{math|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 {{math|ℝ³}}.{{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}}

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 an continuously differentiable isometric embedding into {{math|ℝ^{2''m'' + 1}}}.{{sfnm|1a1=Nash|1y=1954|1pp=394–395}}

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

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{{reflist|30em}}

== References ==

*{{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 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}}

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* {{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 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=15.2|title-link=Foundations of differential geometry|publisher=[[John Wiley & Sons, Inc.]]|___location=New York–London|year=1969|others=Reprinted in 1996|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}}

* {{cite journal|first=Nicolaas H.|last=Kuiper|authorlink=Nicolaas Kuiper|title=On {{math|''C''¹}}-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}}