Content deleted Content added
m →References: refbegin, refend |
No edit summary |
||
Line 19:
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}}
==''C''<sup>''k''</sup> embedding theorem==
Line 32 ⟶ 34:
{{refbegin}}
*{{cite book|last1=Burago|first1=Yu. D.|last2=Zalgaller|first2=V. A.|title=Geometric inequalities|others=Translated from the Russian by A. B. Sosinskiĭ|series=Grundlehren der mathematischen Wissenschaften|volume=285|publisher=[[Springer-Verlag]]|___location=Berlin|year=1988|isbn=3-540-13615-0|mr=0936419|author-link1=Yuri Burago|author-link2=Victor Zalgaller|doi=10.1007/978-3-662-07441-1}}
*{{cite journal|last1=De Lellis|first1=Camillo|last2=Székelyhidi|first2=László, Jr.|title=Dissipative continuous Euler flows|journal=[[Inventiones Mathematicae]]|volume=193|year=2013|issue=2|pages=377–407|mr=3090182|author-link1=Camillo De Lellis|doi=10.1007/s00222-012-0429-9}}
*{{cite book|last1=Eliashberg|first1=Y.|last2=Mishachev|first2=N.|title=Introduction to the h-principle|series=[[Graduate Studies in Mathematics]]|volume=48|publisher=[[American Mathematical Society]]|___location=Providence, RI|year=2002|isbn=0-8218-3227-1|mr=1909245|author-link1=Yakov Eliashberg|doi=10.1090/gsm/048}}
* {{cite journal|last1=Greene|first1=Robert E.|author1-link= Robert Everist Greene |last2 = Jacobowitz|first2=Howard|title= Analytic isometric embeddings|journal=[[Annals of Mathematics]]|series=Second Series|volume=93|pages=189–204|doi=10.2307/1970760|issue=1|year=1971|jstor=1970760|mr=0283728}}
Line 37 ⟶ 40:
* {{cite journal|first=Matthias|last=Günther|title=Zum Einbettungssatz von J. Nash | issue=1|trans-title=On the embedding theorem of J. Nash | language=German |
journal=[[Mathematische Nachrichten]]|volume= 144 |year=1989|pages= 165–187|doi=10.1002/mana.19891440113 | mr=1037168|url = https://onlinelibrary.wiley.com/doi/abs/10.1002/mana.19891440113}}
*{{cite journal|last1=Isett|first1=Philip|title=A proof of Onsager's conjecture|journal=[[Annals of Mathematics]]|series=Second Series|year=2018|issue=3|pages=871–963|mr=3866888|doi=10.4007/annals.2018.188.3.4}}
*{{cite book|mr=0238225|last1=Kobayashi|first1=Shoshichi|last2=Nomizu|first2=Katsumi|author-link2=Katsumi Nomizu|title=Foundations of differential geometry. Vol II|series=Interscience Tracts in Pure and Applied Mathematics|volume=15.2|title-link=Foundations of differential geometry|publisher=[[John Wiley & Sons, Inc.]]|___location=New York–London|year=1969|others=Reprinted in 1996|isbn=0-471-15732-5|author-link1=Shoshichi Kobayashi}}
* {{cite journal|first=Nicolaas H.|last=Kuiper|authorlink=Nicolaas Kuiper|title=On {{math|''C''<sup>1</sup>}}-isometric imbeddings. I|journal=[[Indagationes Mathematicae|Indagationes Mathematicae (Proceedings)]]|volume=58|year=1955a|pages=545–556|mr=0075640|doi=10.1016/S1385-7258(55)50075-8}}
* {{cite journal|first=Nicolaas H.|last=Kuiper|authorlink=Nicolaas Kuiper|title=On {{math|''C''<sup>1</sup>}}-isometric imbeddings. II|journal=[[Indagationes Mathematicae|Indagationes Mathematicae (Proceedings)]]|volume=58|year=1955b|pages=683–689|mr=0075640|doi=10.1016/S1385-7258(55)50093-X}}
*{{cite journal|last1=Müller|first1=S.|last2=Šverák|first2=V.|title=Convex integration for Lipschitz mappings and counterexamples to regularity|journal=[[Annals of Mathematics]]|series=Second Series|volume=157|year=2003|issue=3|pages=715–742|mr=1983780|author-link1=Stefan Müller (mathematician)|author-link2=Vladimir Šverák|doi=10.4007/annals.2003.157.715}}
* {{cite journal|first=John|last=Nash|authorlink=John Forbes Nash, Jr.|title={{math|''C''<sup>1</sup>}} isometric imbeddings|journal=[[Annals of Mathematics]]|series=Second Series|volume=60|year=1954|pages=383–396|doi=10.2307/1969840|issue=3|jstor=1969840|mr=0065993}}
* {{wikicite|ref={{sfnRef|Nash|1956}}|reference={{cite journal|first=John|last=Nash|authorlink=John Forbes Nash, Jr.|title=The imbedding problem for Riemannian manifolds|journal=[[Annals of Mathematics]]|series=Second Series|volume=63|year=1956|pages=20–63|doi=10.2307/1969989|issue=1|mr=0075639|jstor=1969989|ref=none}} {{erratum|https://web.math.princeton.edu/jfnj/texts_and_graphics/Main.Content/Erratum.txt|checked=yes}}}}
| |||