Nash embedding theorems: Difference between revisions

Revision as of 22:19, 19 February 2022

Revision as of 22:23, 19 February 2022

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for all vectors ''u'', ''v'' in ''T_pM''. When {{mvar|n}} is larger than {{math|{{sfrac|1|2}}''m''(''m'' + 1)}}, this is an underdetermined 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 [[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 '''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 prove the existence of aconstruct 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. There is also an older method called [[Kantorovich theorem|Kantorovich iteration]] that uses Newton's method directly (without the introduction of smoothing operators).

== Notes ==