Isothermal coordinates

In mathematics, specifically in differential geometry, isothermal coordinates on a Riemannian manifold are local coordinates where the metric is conformal to the Euclidean metric. This means that in isothermal coordinates, the Riemannian metric locally has the form

$g=\varphi (dx_{1}^{2}+\cdots +dx_{n}^{2}),$

where $\varphi$ is a positive smooth function. (If the Riemannian manifold is oriented, some authors insist that a coordinate system must agree with that orientation to be isothermal.)

Isothermal coordinates on surfaces were first introduced by Gauss. Korn and Lichtenstein proved that isothermal coordinates exist around any point on a two dimensional Riemannian manifold.

By contrast, most higher-dimensional manifolds do not admit isothermal coordinates anywhere; that is, they are not usually locally conformally flat. In dimension 3, a Riemannian metric is locally conformally flat if and only if its Cotton tensor vanishes. In dimensions > 3, a metric is locally conformally flat if and only if its Weyl tensor vanishes.

Isothermal coordinates on surfaces

Gauss (1825) proved the existence of isothermal coordinates on an arbitrary surface with a real analytic metric, following results of Lagrange (1779) on surfaces of revolution. Results for Hölder continuous metrics were obtained by Korn (1916) and Lichtenstein (1916). Later accounts were given by Morrey (1938), Ahlfors (1955), Bers (1952) and Chern (1955). A particularly simple account using the Hodge star operator is given in DeTurck & Kazdan (1981).

Beltrami equation

The existence of isothermal coordinates can be proved by applying known existence theorems for the Beltrami equation, which rely on L^p estimates for singular integral operators of Calderón and Zygmund. A simpler approach to the Beltrami equation has been given more recently by Adrien Douady.

If the Riemannian metric is given locally as

$ds^{2}=E\,dx^{2}+2F\,dx\,dy+G\,dy^{2},$

then in the complex coordinate $z=x+iy$, it takes the form

$ds^{2}=\lambda |\,dz+\mu \,d{\overline {z}}|^{2},$

where $\lambda$ and $\mu$ are smooth with $\lambda >0$ and $\left\vert \mu \right\vert <1$. In fact

$\lambda ={1 \over 4}(E+G+2{\sqrt {EG-F^{2}}}),\,\,\,{\displaystyle \mu ={(E-G+2iF) \over 4\lambda }}.$

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

$ds^{2}=e^{\rho }(du^{2}+dv^{2})$

with ρ smooth. The complex coordinate $w=u+iv$ satisfies

$e^{\rho }\,|dw|^{2}=e^{\rho }|w_{z}|^{2}|\,dz+{w_{\overline {z}} \over w_{z}}\,d{\overline {z}}|^{2},$

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

${\partial w \over \partial {\overline {z}}}=\mu {\partial w \over \partial z}$

has a diffeomorphic solution. Such a solution has been proved to exist in any neighbourhood where $\lVert \mu \rVert _{\infty }<1$.

Existence via local solvability for elliptic partial differential equations

The existence of isothermal coordinates on a smooth two-dimensional Riemannian manifold is a corollary of the standard local solvability result in the analysis of elliptic partial differential equations. In the present context, the relevant elliptic equation is the condition for a function to be harmonic relative to the Riemannian metric. The local solvability then states that any point p has a neighborhood U on which there is a harmonic function u with nowhere-vanishing derivative.

Isothermal coordinates are constructed from such a function in the following way. Harmonicity of u is identical to the closedness of the differential 1-form $\star du,$ defined using the Hodge star operator $\star$ associated to the Riemannian metric. The Poincaré lemma thus implies the existence of a function v on U with $dv=\star du.$ By definition of the Hodge star, $du$ and $dv$ are orthogonal to one another and hence linearly independent, and it then follows from the inverse function theorem that u and v form a coordinate system on some neighborhood of p. This coordinate system is automatically isothermal, since the orthogonality of $du$ and $dv$ implies the diagonality of the metric, and the norm-preserving property of the Hodge star implies the equality of the two diagonal components.

Gaussian curvature

In the isothermal coordinates $(u,v)$, the Gaussian curvature takes the simpler form

$K=-{\frac {1}{2}}e^{-\rho }\left({\frac {\partial ^{2}\rho }{\partial u^{2}}}+{\frac {\partial ^{2}\rho }{\partial v^{2}}}\right).$

See also

• Conformal map
• Liouville's equation
• Quasiconformal map

References

• Ahlfors, Lars V. (1952), "Conformality with respect to Riemannian metrics.", Ann. Acad. Sci. Fenn. Ser. A. I, 206: 1–22
• Ahlfors, Lars V. (1966), Lectures on quasiconformal mappings, Van Nostrand
• Bers, Lipman (1958). Riemann surfaces. Notes taken by Rodlitz, Esther and Pollack, Richard. Courant Institute of Mathematical Sciences at New York University. pp. 15–35.
• Bers, Lipman; John, Fritz; Schechter, Martin (1979). Partial differential equations. Lectures in Applied Mathematics. Vol. 3A. American Mathematical Society. ISBN 0-8218-0049-3.
• Chern, Shiing-shen (1955). "An elementary proof of the existence of isothermal parameters on a surface". Proceedings of the American Mathematical Society. 6 (5): 771–782. doi:10.2307/2032933. JSTOR 2032933.
• DeTurck, Dennis M.; Kazdan, Jerry L. (1981). "Some regularity theorems in Riemannian geometry". Annales Scientifiques de l'École Normale Supérieure. Série 4. 14 (3): 249–260. doi:10.24033/asens.1405. ISSN 0012-9593. MR 0644518..
• do Carmo, Manfredo (1976). Differential Geometry of Curves and Surfaces. Prentice Hall. ISBN 0-13-212589-7.
• Douady, Adrien; Buff, X. (2000), Le théorème d'intégrabilité des structures presque complexes. [Integrability theorem for almost complex structures], London Mathematical Society Lecture Note Series, vol. 274, Cambridge University Press, pp. 307–324

• Gauss, C. F. (1825). "Allgemeine Auflösung der Aufgabe die Theile einer gegebenen Flache auf einer andern gegebnen Fläche so abzubilden, dass die Abbildung dem Abgebildeten in den kleinsten Theilen ähnlich wird" [General solution of the problem of mapping the parts of a given surface on another given surface in such a way that the mapping resembles what is depicted in the smallest parts]. In Schumacher, H. C. (ed.). Astronomische Abhandlungen, Drittes Heft. Altona: Hammerich und Heineking. pp. 1–30. Reprinted in:
  • Gauss, Carl Friedrich (2011) [1873]. Werke: Volume 4. Cambridge Library Collection (in German). New York: Cambridge University Press. doi:10.1017/CBO9781139058254.005. ISBN 978-1-108-03226-1.
  Translated to English in:
  • Gauss (1929). "On conformal representation". In Smith, David Eugene (ed.). A source book in mathematics. Source Books in the History of the Sciences. Translated by Evans, Herbert P. New York: McGraw-Hill Book Co. pp. 463–475. JFM 55.0583.01.

• Imayoshi, Y.; Taniguchi, M. (1992). An introduction to Teichmüller spaces. Tokyo: Springer-Verlag. doi:10.1007/978-4-431-68174-8. ISBN 0-387-70088-9. MR 1215481. Zbl 0754.30001.
• Korn, A. (1914). "Zwei Anwendungen der Methode der sukzessiven Annäherungen". In Carathéodory, C.; Hessenberg, G.; Landau, E.; Lichtenstein, L. (eds.). Mathematische Abhandlungen Hermann Amandus Schwarz. Berlin, Heidelberg: Springer. pp. 215–229. doi:10.1007/978-3-642-50735-9_16. ISBN 978-3-642-50426-6.
• Lagrange, J. (1779). "Sur la construction des cartes géographiques". Nouveaux mémoires de l'Académie royale des sciences et belles-lettres de Berlin: 161–210. Reprinted in:
  • Serret, J.-A., ed. (1867). Œuvres de Lagrange: tome 4 (in French). Paris: Gauthier-Villars.
• Lichtenstein, Léon (1916). "Zur Theorie der konformen Abbildung. Konforme Abbildung nichtanalytischer, singularitätenfreier Flächenstücke auf ebene Gebiete". Bulletin International de l'Académie des Sciences de Cracovie: Classe des Sciences Mathématiques et Naturelles. Série A: Sciences Mathématiques: 192–217. JFM 46.0547.01.
• Morrey, Charles B. (1938). "On the solutions of quasi-linear elliptic partial differential equations". Trans. Amer. Math. Soc. 43 (1). American Mathematical Society: 126–166. doi:10.2307/1989904. JSTOR 1989904.
• Spivak, Michael (1999). A comprehensive introduction to differential geometry. Volume four (Third edition of 1975 original ed.). Publish or Perish, Inc. ISBN 0-914098-73-X. MR 0532833. Zbl 1213.53001.
• Taylor, Michael E. (2000). Tools for PDE. Pseudodifferential operators, paradifferential operators, and layer potentials. Mathematical Surveys and Monographs. Vol. 81. Providence, RI: American Mathematical Society. doi:10.1090/surv/081. ISBN 0-8218-2633-6. MR 1766415. Zbl 0963.35211.
• Taylor, Michael E. (2011). Partial differential equations I. Basic theory. Applied Mathematical Sciences. Vol. 115 (Second edition of 1996 original ed.). New York: Springer. doi:10.1007/978-1-4419-7055-8. ISBN 978-1-4419-7054-1. MR 2744150. Zbl 1206.35002.

External links

• "Isothermal coordinates", Encyclopedia of Mathematics, EMS Press, 2001 [1994]