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The theorem was originally proved by John Nash with the condition ''n'' ≥ ''m''+2 instead of ''n'' ≥ ''m''+1 and generalized by [[Nicolaas Kuiper]], by a relatively easy trick.
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
==''C''<sup>''k''</sup> embedding theorem==
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