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Any closed and oriented two-dimensional manifold can be smoothly embedded in {{math|ℝ<sup>3</sup>}}. Any such embedding can be scaled by an arbitrarily small constant so as to become short, relative to any given Riemannian metric on the surface. It follows from the Nash–Kuiper theorem that there are continuously differentiable isometric embeddings of any such Riemannian surface where the radius of a circumscribed ball is arbitrarily small. By contrast, no negatively curved closed surface can even be smoothly isometrically embedded in {{math|ℝ<sup>3</sup>}}.{{sfnm|1a1=Kobayashi|1a2=Nomizu|1y=1969|1loc=Theorem VII.5.6}} Moreover, for any smooth (or even {{math|''C''<sup>2</sup>}}) isometric embedding of an arbitrary closed Riemannian surface, there is a quantitative (positive) lower bound on the radius of a circumscribed ball in terms of the surface area and curvature of the embedded metric.{{sfnm|1a1=Burago|1a2=Zalgaller|1y=1988|1loc=Corollary 6.2.2}}
In higher dimension, as follows from the [[Whitney embedding theorem]], the Nash–Kuiper theorem shows that any closed {{mvar|m}}-dimensional Riemannian manifold admits an continuously differentiable isometric embedding into an ''arbitrarily small neighborhood'' in {{math|2''m''}}-dimensional Euclidean space. Although Whitney's theorem also applies to noncompact manifolds, such embeddings cannot simply be scaled by a small constant so as to become short. Nash proved that every {{mvar|m}}-dimensional Riemannian manifold admits
==''C''<sup>''k''</sup> embedding theorem==
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