Nash embedding theorems: Difference between revisions

Revision as of 06:13, 6 September 2021

Revision as of 11:48, 7 September 2021

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==''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, or ''n'' ≤ ''m''(''m''+1)(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 1956-01|title=The Imbedding Problem for Riemannian Manifolds|url=https://www.jstor.org/stable/1969989?origin=crossref|journal=The Annals of Mathematics|volume=63|issue=1|pages=20|doi=10.2307/1969989}}</ref> That is ƒ is an [[Embedding#Differential_topologyDifferential 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:

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

for all vectors ''u'', ''v'' in ''T_pM''. This is an undetermined system of [[partial differential equation]]s (PDEs).

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The Nash embedding theorem is a global theorem in the sense that the whole manifold is embedded into '''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-reaching generalization of the implicit function theorem, the [[Nash–Moser theorem]] and Newton's method with postconditioning. The basic idea of Nash's solution of the embedding problem is the use of [[Newton's method]] to prove the existence of a solution to the above system of PDEs. 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. There is also an older method called [[Kantorovich theorem|Kantorovich iteration]] that uses Newton's method directly (without the introduction of smoothing operators).

==References==

{{Reflist}}

* {{citation|last=Greene|first=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}}

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