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→Nash–Kuiper theorem ({{math|C1}} embedding theorem) {{anchor|Nash–Kuiper theorem}}: less subjective phrasing (which in my opinion is also more correct) |
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In particular, as follows from the [[Whitney embedding theorem]], any ''m''-dimensional Riemannian manifold admits an isometric ''C''<sup>1</sup>-embedding into an ''arbitrarily small neighborhood'' in 2''m''-dimensional Euclidean space.
The theorem was originally proved by John Nash with the condition {{math|''n'' ≥ ''m'' + 2}}.
The theorem has many counterintuitive implications. For example, it follows that any closed oriented Riemannian surface can be ''C''<sup>1</sup> isometrically embedded into an arbitrarily small [[ball (mathematics)|ε-ball]] in Euclidean 3-space (for small <math>\epsilon</math> there is no such ''C''<sup>2</sup>-embedding since from the [[Gaussian curvature#Alternative formulas|formula for the Gauss curvature]] an extremal point of such an embedding would have curvature ≥ ε<sup>−2</sup>). And, there exist ''C''<sup>1</sup> isometric embeddings of the hyperbolic plane in '''R'''<sup>3</sup>.
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