Nash embedding theorems: Difference between revisions

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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 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]].

==See also==

* {{annotated link|Representation theorem}}

* {{annotated link|Whitney immersion theorem}}

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

* {{annotated link|Nonlinear dimensionality reduction}}

* {{annotated link|Universal space}}

== Citations ==