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| In [[mathematics]], specifically in [[differential geometry]], '''isothermal coordinates''' on a [[Riemannian manifold]]
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| are local coordinates where the [[metric tensor|metric]] is
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| [[Conformal geometry|conformal]] to the [[Euclidean metric]]. This means that in isothermal
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| coordinates, the [[Riemannian metric]] locally has the form
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| :<math> g = e^\varphi (dx_1^2 + \cdots + dx_n^2),</math>
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| where <math>\varphi</math> is a [[smooth function]].
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| Isothermal coordinates on surfaces were first introduced by [[Carl Friedrich Gauss|Gauss]]. Korn and Lichtenstein proved that isothermal coordinates exist around any point on a two dimensional Riemannian manifold. On higher dimensional Riemannian manifolds a necessary and sufficient condition for their local existence is the vanishing of the [[Weyl tensor]] and of the [[Cotton tensor]].
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| ==Isothermal coordinates on surfaces==
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| {{harvtxt|Gauss|1822}} proved the existence of isothermal coordinates on an arbitrary surface with a real analytic metric, following results of
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| {{harvtxt|Lagrange|1779}} on surfaces of revolution. Results for Hölder continuous metrics were obtained by {{harvtxt|Korn|1916}} and {{harvtxt|Lichtenstein|1916}}. Later accounts were given by {{harvtxt|Morrey|1938}}, {{harvtxt|Ahlfors|1955}}, {{harvtxt|Bers|1952}} and {{harvtxt|Chern|1955}}. A particularly simple account using the [[Hodge star operator]] is given in {{harvtxt|DeTurck|Kazdan|1981}}.
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| ===Beltrami equation===
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| The existence of isothermal coordinates can be proved<ref>{{harvnb|Imayoshi|Taniguchi|1992|pp=20–21}}</ref> by applying known existence theorems for the [[Beltrami equation]], which rely on L<sup>p</sup> estimates for [[singular integral operator]]s of [[Alberto Calderon|Calderon]] and [[Antoni Zygmund|Zygmund]].<ref>{{harvnb|Ahlfors|1966|pp=85–115}}</ref><ref>{{harvnb|Imayoshi|Taniguchi|1992|pp=92–104}}</ref> A simpler approach to the Beltrami equation has been given more recently by the late [[Adrien Douady]].<ref>{{harvnb|Douady|Buff|2000}}</ref> | |
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| If the Riemannian metric is given locally as
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| :<math> ds^2 = E \, dx^2 + 2F \, dx \, dy + G \, dy^2,</math>
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| then in the complex coordinate ''z'' = ''x'' + i''y'', it takes the form
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| :<math> ds^2 = \lambda| \, dz +\mu \, d\overline{z}|^2,</math>
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| where λ and μ are smooth with λ > 0 and |μ| < 1. In fact
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| :<math> \lambda={1\over 4} ( E + G +2\sqrt{EG -F^2}),\,\,\, \mu=(E - G + 2iF)/4\lambda.</math>
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| In isothermal coordinates (''u'', ''v'') the metric should take the form
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| :<math> ds^2 = \rho (du^2 + dv^2)</math>
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| with ρ > 0 smooth. The complex coordinate ''w'' = ''u'' + i ''v'' satisfies
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| :<math>\rho \, |dw|^2 = \rho |w_{z}|^2 | \, dz + {w_{\overline {z}}\over w_z} \, d\overline{z}|^2,</math>
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| so that the coordinates (''u'', ''v'') will be isothermal if the '''Beltrami equation'''
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| :<math> {\partial w\over \partial \overline{z}} = \mu {\partial w\over \partial z}</math>
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| has a diffeomorphic solution. Such a solution has been proved to exist in any neighbourhood where ||μ||<sub>∞</sub> < 1.
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| ===Hodge star operator===
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| New coordinates ''u'' and ''v'' are isothermal provided that
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| :<math> \star du =dv,</math>
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| where <math>\star</math> is the [[Hodge star operator]] defined by the metric.<ref>{{harvnb|DeTurck|Kazdan|1981}}; {{harvnb|Taylor|1996|pp=377–378}}</ref>
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| Let <math> \Delta=d^*d</math> be the [[Laplace–Beltrami operator]] on functions.
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| Then by standard elliptic theory, ''u'' can be chosen to be [[harmonic]] near a given point, i.e. Δ ''u'' = 0, with ''du'' non-vanishing.
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| By the [[Poincaré lemma]] <math>\star du=dv</math> has a local solution ''v'' exactly when <math>d\star d u =0</math>.
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| Since
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| :<math>\star d \star = d^*,</math>
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| this is equivalent to Δ ''u'' = 0, and hence a local solution exists.
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| Since ''du'' is non-zero and the square of the Hodge star operator is −1 on 1-forms, ''du'' and ''dv'' are necessarily linearly independent, and therefore give local isothermal coordinates.
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| ===Gaussian curvature===
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| In the isothermal coordinates (''u'', ''v''), the [[Gaussian curvature]] takes the simpler form
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| : <math> K = -\frac{1}{2} e^{-\varphi} \left(\frac{\partial^2 \varphi}{\partial u^2} + \frac{\partial^2 \varphi}{\partial v^2}\right),</math> | |
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| where <math> \rho = e^\varphi</math>.
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| ==See also==
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| *[[Conformal map]]
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| *[[Liouville's equation]]
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| *[[Quasiconformal map]]
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| ==Notes==
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| {{reflist}}
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| ==References==
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| * {{citation|last=Ahlfors|first=Lars V.|authorlink=Lars Ahlfors|title=Conformality with respect to Riemannian metrics.|series=Ann. Acad. Sci. Fenn. Ser. A. I.|year=1952|volume= 206|pages=1–22}}
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| * {{citation|last=Ahlfors|first=Lars V.|authorlink=Lars Ahlfors|title=Lectures on quasiconformal mappings|publisher=Van Nostrand|year=1966}}
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| * {{citation|last=Bers|first=Lipman|authorlink=Lipman Bers|title=Riemann Surfaces, 1951–1952|publisher=New York University|year=1952|pages=15–35}}
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| * {{citation|first= Shiing-shen|last=Chern|authorlink=S. S. Chern|title=An elementary proof of the existence of isothermal parameters on a surface|
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| journal=Proc. Amer. Math. Soc.|volume= 6 |year=1955|pages= 771–782|doi= 10.2307/2032933|jstor= 2032933|issue= 5|publisher= American Mathematical Society}}
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| * {{Citation | last1=DeTurck | first1=Dennis M. | last2=Kazdan | first2=Jerry L. | author2-link=Jerry Kazdan | title=Some regularity theorems in Riemannian geometry | url=http://www.numdam.org/item?id=ASENS_1981_4_14_3_249_0 | id={{MathSciNet | id = 644518}} | year=1981 | journal=Annales Scientifiques de l'École Normale Supérieure. Quatrième Série | issn=0012-9593 | volume=14 | issue=3 | pages=249–260}}.
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| * {{citation|first=Manfredo |last=do Carmo| title=Differential Geometry of Curves and Surfaces|publisher=Prentice Hall|year=1976|id=ISBN 0-13-212589-7}}
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| *{{citation|last=Douady|first= Adrien|authorlink=Adrien Douady|last2= Buff|first2= X.|title=Le théorème d'intégrabilité des structures presque complexes. [Integrability theorem for almost complex structures]|pages= 307–324|
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| series=London Math. Soc. Lecture Note Ser.|volume= 274|year= 2000|publisher =Cambridge Univ. Press}}
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| *{{citation|first=C.F.|last=Gauss|title=On Conformal Representation|year=1822|translator=Smith, Eugene|url=http://archive.org/details/sourcebookinmath00smit|pages=463-475}}
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| *{{citation|first=Y. |last=Imayoshi|first2=M.|last2=Taniguchi|title=An Introduction to Teichmüller spaces|publisher=Springer-Verlag|year=1992|id=ISBN 0-387-70088-9}}
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| *{{citation|first=A.|last=Korn|title=Zwei Anwendungen der Methode der sukzessiven Annäherungen|series=Schwarz Abhandlungen|year=1916|pages=215–219}}
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| *{{citation|first=J.|last= Lagrange|title=Sur la construction des cartes géographiques|year=1779|url=http://math-doc.ujf-grenoble.fr/cgi-bin/oeitem?id=OE_LAGRANGE__4_637_0}}
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| *{{citation|first=L.|last= Lichtenstein|title=Zur Theorie der konformen Abbildung
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| |journal= Bull. Internat. Acad. Sci. Cracovie. Cl. Sci. Math. Nat. Sér. A.|year= 1916|pages= 192–217}}
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| *{{citation|first=Charles B.|last=Morrey|authorlink=Charles B. Morrey, Jr.|title=On the solutions of quasi-linear elliptic partial differential equations|journal=Trans. Amer. Math. Soc.|year=1938|pages=126–166|doi=10.2307/1989904|volume=43|jstor=1989904|issue=1|publisher=American Mathematical Society}}
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| *{{citation|first=Michael|last= Spivak|authorlink=Michael Spivak|title=A Comprehensive Introduction to Differential Geometry, 3rd edition| publisher= Publish or Perish}}
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| *{{citation|first=Michael E.|last=Taylor|authorlink=Michael E. Taylor|title=Partial Differential Equations: Basic Theory|publisher=Springer-Verlag|year=1996|id=ISBN 0-387-94654-3|
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| pages=376–378}}
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| ==External links==
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| * {{springer|title=Isothermal coordinates|id=p/i052890}}
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| [[Category:Differential geometry]]
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| [[Category:Coordinate systems in differential geometry]]
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| [[Category:Partial differential equations]]
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Friends call him Royal Seyler. The thing I adore most bottle tops gathering and now I have time to take on new issues. Interviewing is what she does in her day job but quickly her spouse and her will begin their own business. Alabama has always been his house and his family loves it.
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