Nash embedding theorems: Difference between revisions

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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=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 [[embeddingEmbedding#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).