Nash embedding theorems: Difference between revisions

Revision as of 20:26, 19 February 2022

Revision as of 20:26, 19 February 2022

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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>

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

'''Theorem.'''{{sfnm|1a1=Eliashberg|1a2=Mishachev|1y=2002|1loc=Chapter 21|2a1=Gromov|2y=1986|2loc=Section 2.4.9}} Let (''M'',''g'') be an ''m''-dimensional Riemannian manifold and ƒ: ''M'' → '''R'''^''n'' a [[short map|short]] ''C''^∞-embedding (or [[Immersion (mathematics)|immersion]]) into Euclidean space '''R'''^''n'', where ''n'' ≥ ''m''+1. This embedding is not required to be isometric. Then for arbitrary ε > 0 there is an embedding (or immersion) ƒ_ε: ''M'' → '''R'''^''n'' which is

{{ordered list|type=lower-roman

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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 ''n'' ≥ ''m''+2 instead of ''n'' ≥ ''m''+1 and generalized by [[Nicolaas Kuiper]], by a relatively easy trick.{{sfnm|1a1=Nash|1y=1954}}{{sfnm|1a1=Kuiper|1y=1955a|2a1=Kuiper|2y=1955b}}

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.