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m n >= m + 2 is a stronger assumption than n >= m + 1 |
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<blockquote>'''Nash–Kuiper theorem.'''{{sfnm|1a1=Eliashberg|1a2=Mishachev|1y=2002|1loc=Chapter 21|2a1=Gromov|2y=1986|2loc=Section 2.4.9}} Let {{math|(''M'', ''g'')}} be an {{mvar|m}}-dimensional Riemannian manifold and {{math|''f'': ''M'' → ℝ<sup>''n''</sup>}} a [[short map|short]] smooth embedding (or [[Immersion (mathematics)|immersion]]) into Euclidean space {{math|ℝ<sup>''n''</sup>}}, where {{math|''n'' ≥ ''m'' + 1}}. This map is not required to be isometric. Then there is a sequence of continuously differentiable isometric embeddings (or immersions) {{math|''M'' → ℝ<sup>''n''</sup>}} of {{mvar|g}} which [[uniform convergence|converge uniformly]] to {{mvar|f}}.</blockquote>
The theorem was originally proved by John Nash with the
The isometric embeddings produced by the Nash–Kuiper theorem are often considered counterintuitive and pathological.{{sfnm|1a1=Kobayashi|1a2=Nomizu|1y=1969|1loc=Note 18}} They often fail to be smoothly differentiable. For example, a [[Hilbert's theorem (differential geometry)|well-known theorem]] of [[David Hilbert]] asserts that the [[hyperbolic plane]] cannot be smoothly isometrically immersed into {{math|ℝ<sup>3</sup>}}. Any [[Einstein manifold]] of negative [[scalar curvature]] cannot be smoothly isometrically immersed as a hypersurface,{{sfnm|1a1=Kobayashi|1a2=Nomizu|1y=1969|1loc=Theorem VII.5.3}} and a theorem of [[Shiing-Shen Chern]] and Kuiper even says that any [[closed manifold|closed]] {{mvar|m}}-dimensional manifold of nonpositive [[sectional curvature]] cannot be smoothly isometrically immersed in {{math|ℝ<sup>2''m'' – 1</sup>}}.{{sfnm|1a1=Kobayashi|1a2=Nomizu|1y=1969|1loc=Corollary VII.4.8}} Furthermore, some smooth isometric embeddings exhibit rigidity phenomena which are violated by the largely unrestricted choice of {{mvar|f}} in the Nash–Kuiper theorem. For example, the image of any smooth isometric hypersurface immersion of the round sphere must itself be a round sphere.{{sfnm|1a1=Kobayashi|1a2=Nomizu|1y=1969|1loc=Corollary VII.5.4 and Note 15}} By contrast, the Nash–Kuiper theorem ensures the existence of continuously differentiable isometric hypersurface immersions of the round sphere which are arbitrarily close to (for instance) a topological embedding of the sphere as a small [[ellipsoid]].
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