Positive-definite kernel

2013-12-16T21:18:37Z

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In [[mathematics]], specifically in [[differential geometry]], '''isothermal coordinates''' on a [[Riemannian manifold]]
are local coordinates where the [[metric tensor|metric]] is
[[Conformal geometry|conformal]] to the [[Euclidean metric]]. This means that in isothermal
coordinates, the [[Riemannian metric]] locally has the form
:<math> g = e^\varphi (dx_1^2 + \cdots + dx_n^2),</math>
where <math>\varphi</math> is a [[smooth function]].

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]].

==Isothermal coordinates on surfaces==
{{harvtxt|Gauss|1822}} proved the existence of isothermal coordinates on an arbitrary surface with a real analytic metric, following results of
{{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}}.

===Beltrami equation===
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>

If the Riemannian metric is given locally as

:<math> ds^2 = E \, dx^2 + 2F \, dx \, dy + G \, dy^2,</math>

then in the complex coordinate ''z'' = ''x'' + i''y'', it takes the form

:<math> ds^2 = \lambda| \, dz +\mu \, d\overline{z}|^2,</math>

where λ and μ are smooth with λ > 0 and |μ| < 1. In fact

:<math> \lambda={1\over 4} ( E + G +2\sqrt{EG -F^2}),\,\,\, \mu=(E - G + 2iF)/4\lambda.</math>

In isothermal coordinates (''u'', ''v'') the metric should take the form

:<math> ds^2 = \rho (du^2 + dv^2)</math>

with ρ > 0 smooth. The complex coordinate ''w'' = ''u'' + i ''v'' satisfies

:<math>\rho \, |dw|^2 = \rho |w_{z}|^2 | \, dz + {w_{\overline {z}}\over w_z} \, d\overline{z}|^2,</math>

so that the coordinates (''u'', ''v'') will be isothermal if the '''Beltrami equation'''

:<math> {\partial w\over \partial \overline{z}} = \mu {\partial w\over \partial z}</math>

has a diffeomorphic solution. Such a solution has been proved to exist in any neighbourhood where ||μ||<sub>∞</sub> < 1.

===Hodge star operator===
New coordinates ''u'' and ''v'' are isothermal provided that

:<math> \star du =dv,</math>

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>

Let <math> \Delta=d^*d</math> be the [[Laplace–Beltrami operator]] on functions.

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.

By the [[Poincaré lemma]] <math>\star du=dv</math> has a local solution ''v'' exactly when <math>d\star d u =0</math>.

Since

:<math>\star d \star = d^*,</math>

this is equivalent to Δ ''u'' = 0, and hence a local solution exists.

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.

===Gaussian curvature===
In the isothermal coordinates (''u'', ''v''), the [[Gaussian curvature]] takes the simpler form

: <math> K = -\frac{1}{2} e^{-\varphi} \left(\frac{\partial^2 \varphi}{\partial u^2} + \frac{\partial^2 \varphi}{\partial v^2}\right),</math>

where <math> \rho = e^\varphi</math>.

==See also==
*[[Conformal map]]
*[[Liouville's equation]]
*[[Quasiconformal map]]

==Notes==
{{reflist}}
==References==
* {{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}}
* {{citation|last=Ahlfors|first=Lars V.|authorlink=Lars Ahlfors|title=Lectures on quasiconformal mappings|publisher=Van Nostrand|year=1966}}
* {{citation|last=Bers|first=Lipman|authorlink=Lipman Bers|title=Riemann Surfaces, 1951–1952|publisher=New York University|year=1952|pages=15–35}}
* {{citation|first= Shiing-shen|last=Chern|authorlink=S. S. Chern|title=An elementary proof of the existence of isothermal parameters on a surface|
journal=Proc. Amer. Math. Soc.|volume= 6 |year=1955|pages= 771–782|doi= 10.2307/2032933|jstor= 2032933|issue= 5|publisher= American Mathematical Society}}
* {{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}}.
* {{citation|first=Manfredo |last=do Carmo| title=Differential Geometry of Curves and Surfaces|publisher=Prentice Hall|year=1976|id=ISBN 0-13-212589-7}}
*{{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|
series=London Math. Soc. Lecture Note Ser.|volume= 274|year= 2000|publisher =Cambridge Univ. Press}}
*{{citation|first=C.F.|last=Gauss|title=On Conformal Representation|year=1822|translator=Smith, Eugene|url=http://archive.org/details/sourcebookinmath00smit|pages=463-475}}
*{{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}}
*{{citation|first=A.|last=Korn|title=Zwei Anwendungen der Methode der sukzessiven Annäherungen|series=Schwarz Abhandlungen|year=1916|pages=215–219}}
*{{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}}
*{{citation|first=L.|last= Lichtenstein|title=Zur Theorie der konformen Abbildung
|journal= Bull. Internat. Acad. Sci. Cracovie. Cl. Sci. Math. Nat. Sér. A.|year= 1916|pages= 192–217}}
*{{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}}
*{{citation|first=Michael|last= Spivak|authorlink=Michael Spivak|title=A Comprehensive Introduction to Differential Geometry, 3rd edition| publisher= Publish or Perish}}
*{{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|
pages=376–378}}

==External links==
* {{springer|title=Isothermal coordinates|id=p/i052890}}

[[Category:Differential geometry]]
[[Category:Coordinate systems in differential geometry]]
[[Category:Partial differential equations]]

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Positive-definite kernel