Nash embedding theorems: Difference between revisions

Revision as of 07:13, 11 September 2023

Revision as of 17:36, 8 October 2023

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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, and with ''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'').{{sfnm|1a1=Nash|1y=1956}} That is ƒ is an [[Embedding#Differential 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''. When {{mvar|n}} is larger than {{math|{{sfrac|1|2}}''m''(''m'' + 1)}}, this is an underdetermined system of [[partial differential equation]]s (PDEs).