Editing Nash embedding theorems (section)

==Nash–Kuiper theorem ({{math|''C''<sup>1</sup>}} embedding theorem) ==
Given an {{mvar|m}}-dimensional Riemannian manifold {{math|(''M'', ''g'')}}, an ''isometric embedding'' is a continuously differentiable [[topological embedding]] {{math|''f'': ''M'' → ℝ<sup>''n''</sup>}} such that the [[pullback]] of the Euclidean metric equals {{mvar|g}}. In analytical terms, this may be viewed (relative to a smooth [[coordinate chart]] {{mvar|x}}) as a system of {{math|{{sfrac|1|2}}''m''(''m'' + 1)}} many first-order [[partial differential equation]]s for {{mvar|n}} unknown (real-valued) functions:
:<math>g_{ij}(x)=\sum_{\alpha=1}^n\frac{\partial f^\alpha}{\partial x^i}\frac{\partial f^\alpha}{\partial x^j}.</math>
If {{mvar|n}} is less than {{math|{{sfrac|1|2}}''m''(''m'' + 1)}}, then there are more equations than unknowns. From this perspective, the existence of isometric embeddings given by the following theorem is considered surprising.
<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 stronger assumption {{math|''n'' ≥ ''m'' + 2}}. His method was modified by [[Nicolaas Kuiper]] to obtain the theorem above.{{sfnm|1a1=Nash|1y=1954}}{{sfnm|1a1=Kuiper|1y=1955a|2a1=Kuiper|2y=1955b}}

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]].

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 a continuously differentiable isometric embedding into {{math|ℝ<sup>2''m'' + 1</sup>}}.{{sfnm|1a1=Nash|1y=1954|1pp=394–395}}

At the time of Nash's work, his theorem was considered to be something of a mathematical curiosity. The result itself has not found major applications. However, Nash's method of proof was adapted by [[Camillo De Lellis]] and László Székelyhidi to construct low-regularity solutions, with prescribed [[kinetic energy]], of the [[Euler equation]]s from the mathematical study of [[fluid mechanics]]. In analytical terms, the Euler equations have a formal similarity to the isometric embedding equations, via the quadratic nonlinearity in the first derivatives of the unknown function.{{sfnm|1a1=De Lellis|1a2=Székelyhidi|1y=2013|2a1=Isett|2y=2018}} The ideas of Nash's proof were abstracted by [[Mikhael Gromov (mathematician)|Mikhael Gromov]] to the principle of ''convex integration'', with a corresponding [[h-principle]].{{sfnm|1a1=Gromov|1y=1986|1loc=Section 2.4}} This was applied by [[Stefan Müller (mathematician)|Stefan Müller]] and [[Vladimír Šverák]] to [[Hilbert's nineteenth problem]], constructing minimizers of minimal differentiability in the [[calculus of variations]].{{sfnm|1a1=Müller|1a2=Šverák|1y=2003}}