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==Nash–Kuiper theorem (''C''<sup>1</sup> embedding theorem) {{anchor|Nash–Kuiper theorem}}==
'''Theorem.''' Let (''M'',''g'') be an ''m''-dimensional Riemannian manifold and ƒ: ''M'' → '''R'''<sup>''n''</sup> a [[short map|short]] ''C''<sup>∞</sup>-embedding (or [[Immersion (mathematics)|immersion]]) into Euclidean space '''R'''<sup>''n''</sup>, where ''n'' ≥ ''m''+1. This embedding is not required to be isometric. Then for arbitrary ε > 0 there is an embedding (or immersion) ƒ<sub>ε</sub>: ''M'' → '''R'''<sup>''n''</sup> which is
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