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In
mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, specifically in
differential geometry Differential geometry is a mathematical discipline that studies the geometry of smooth shapes and smooth spaces, otherwise known as smooth manifolds. It uses the techniques of differential calculus, integral calculus, linear algebra and mult ...
, isothermal coordinates on a
Riemannian manifold In differential geometry, a Riemannian manifold or Riemannian space , so called after the German mathematician Bernhard Riemann, is a real, smooth manifold ''M'' equipped with a positive-definite inner product ''g'p'' on the tangent space ...
are local coordinates where the
metric Metric or metrical may refer to: * Metric system, an internationally adopted decimal system of measurement * An adjective indicating relation to measurement in general, or a noun describing a specific type of measurement Mathematics In mathe ...
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 In mathematical analysis, the smoothness of a function is a property measured by the number of continuous derivatives it has over some domain, called ''differentiability class''. At the very minimum, a function could be considered smooth if ...
. (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 In differential geometry, the Cotton tensor on a (pseudo)- Riemannian manifold of dimension ''n'' is a third-order tensor concomitant of the metric. The vanishing of the Cotton tensor for is necessary and sufficient condition for the manifol ...
vanishes. In dimensions > 3, a metric is locally conformally flat if and only if its
Weyl tensor In differential geometry, the Weyl curvature tensor, named after Hermann Weyl, is a measure of the curvature of spacetime or, more generally, a pseudo-Riemannian manifold. Like the Riemann curvature tensor, the Weyl tensor expresses the tida ...
vanishes.


Isothermal coordinates on surfaces

In 1822,
Carl Friedrich Gauss Johann Carl Friedrich Gauss (; german: Gauß ; la, Carolus Fridericus Gauss; 30 April 177723 February 1855) was a German mathematician and physicist who made significant contributions to many fields in mathematics and science. Sometimes refer ...
proved the existence of isothermal coordinates on an arbitrary surface with a real-analytic Riemannian metric, following earlier results of
Joseph Lagrange Joseph-Louis Lagrange (born Giuseppe Luigi Lagrangiasurfaces of revolution A surface of revolution is a surface in Euclidean space created by rotating a curve (the generatrix) around an axis of rotation. Examples of surfaces of revolution generated by a straight line are cylindrical and conical surfaces depending on ...
. The construction used by Gauss made use of the
Cauchy–Kowalevski theorem In mathematics, the Cauchy–Kovalevskaya theorem (also written as the Cauchy–Kowalevski theorem) is the main local existence and uniqueness theorem for analytic partial differential equations associated with Cauchy initial value problems. A ...
, so that his method is fundamentally restricted to the real-analytic context. Following innovations in the theory of two-dimensional
partial differential equations In mathematics, a partial differential equation (PDE) is an equation which imposes relations between the various partial derivatives of a multivariable function. The function is often thought of as an "unknown" to be solved for, similarly to ...
by
Arthur Korn Arthur Korn (20 May 1870 – 21 December/22 December 1945) was a German physicist, mathematician and inventor. He was involved in the development of the fax machine, specifically the transmission of photographs or telephotography, known as the ...
,
Leon Lichtenstein Leon Lichtenstein (16 May 1878 – 21 August 1933) was a Polish-German mathematician, who made contributions to the areas of differential equations, conformal mapping, and potential theory. He was also interested in theoretical physics, publis ...
found in 1916 the general existence of isothermal coordinates for Riemannian metrics of lower regularity, including smooth metrics and even
Hölder continuous Hölder: * ''Hölder, Hoelder'' as surname * Hölder condition * Hölder's inequality * Hölder mean * Jordan–Hölder theorem In abstract algebra, a composition series provides a way to break up an algebraic structure, such as a group or a modu ...
metrics. Given a Riemannian metric on a two-dimensional manifold, the transition function between isothermal coordinate charts, which is a map between open subsets of , is necessarily angle-preserving. The angle-preserving property together with orientation-preservation is one characterization (among many) of
holomorphic function In mathematics, a holomorphic function is a complex-valued function of one or more complex variables that is complex differentiable in a neighbourhood of each point in a domain in complex coordinate space . The existence of a complex deriv ...
s, and so an ''oriented'' coordinate atlas consisting of isothermal coordinate charts may be viewed as a holomorphic coordinate atlas. This demonstrates that a Riemannian metric and an orientation on a two-dimensional manifold combine to induce the structure of a
Riemann surface In mathematics, particularly in complex analysis, a Riemann surface is a connected one-dimensional complex manifold. These surfaces were first studied by and are named after Bernhard Riemann. Riemann surfaces can be thought of as deformed ver ...
(i.e. a one-dimensional
complex manifold In differential geometry and complex geometry, a complex manifold is a manifold with an atlas of charts to the open unit disc in \mathbb^n, such that the transition maps are holomorphic. The term complex manifold is variously used to mean a ...
). Furthermore, given an oriented surface, two Riemannian metrics induce the same holomorphic atlas if and only if they are conformal to one another. For this reason, the study of Riemann surfaces is identical to the study of conformal classes of Riemannian metrics on oriented surfaces. By the 1950s, expositions of the ideas of Korn and Lichtenstein were put into the language of
complex derivative In mathematics, a holomorphic function is a complex-valued function of one or more complex variables that is complex differentiable in a neighbourhood of each point in a domain in complex coordinate space . The existence of a complex deriv ...
s and the
Beltrami equation In mathematics, the Beltrami equation, named after Eugenio Beltrami, is the partial differential equation : = \mu . for ''w'' a complex distribution of the complex variable ''z'' in some open set ''U'', with derivatives that are locally ''L'' ...
by
Lipman Bers Lipman Bers ( Latvian: ''Lipmans Berss''; May 22, 1914 – October 29, 1993) was a Latvian-American mathematician, born in Riga, who created the theory of pseudoanalytic functions and worked on Riemann surfaces and Kleinian groups. He was also ...
and Shiing-shen Chern, among others. In this context, it is natural to investigate the existence of generalized solutions, which satisfy the relevant partial differential equations but are no longer interpretable as coordinate charts in the usual way. This was initiated by
Charles Morrey Charles Bradfield Morrey Jr. (July 23, 1907 – April 29, 1984) was an American mathematician who made fundamental contributions to the calculus of variations and the theory of partial differential equations. Life Charles Bradfield Morrey Jr. ...
in his seminal 1938 article on the theory of
elliptic partial differential equation Second-order linear partial differential equations (PDEs) are classified as either elliptic, hyperbolic, or parabolic. Any second-order linear PDE in two variables can be written in the form :Au_ + 2Bu_ + Cu_ + Du_x + Eu_y + Fu +G= 0,\, whe ...
s on two-dimensional domains, leading later to the measurable Riemann mapping theorem of
Lars Ahlfors Lars Valerian Ahlfors (18 April 1907 – 11 October 1996) was a Finnish mathematician, remembered for his work in the field of Riemann surfaces and his text on complex analysis. Background Ahlfors was born in Helsinki, Finland. His mother, Si ...
and Bers.


Beltrami equation

The existence of isothermal coordinates can be proved by applying known existence theorems for the
Beltrami equation In mathematics, the Beltrami equation, named after Eugenio Beltrami, is the partial differential equation : = \mu . for ''w'' a complex distribution of the complex variable ''z'' in some open set ''U'', with derivatives that are locally ''L'' ...
, which rely on Lp estimates for
singular integral operator In mathematics, singular integrals are central to harmonic analysis and are intimately connected with the study of partial differential equations. Broadly speaking a singular integral is an integral operator : T(f)(x) = \int K(x,y)f(y) \, dy, wh ...
s 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, ^2, where \lambda and \mu are smooth with \lambda>0 and \left\vert \mu \right\vert < 1. In fact : \lambda= ( E + G +2\sqrt),\,\,\, . In isothermal coordinates (u,v) the metric should take the form : ds^2 = e^ (du^2 + dv^2) with ρ smooth. The complex coordinate w = u + iv satisfies :e^ \, , dw, ^2 = e^ , w_, ^2 , \, dz + \, d\overline, ^2, so that the coordinates (''u'', ''v'') will be isothermal if the Beltrami equation : = \mu 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 equation Second-order linear partial differential equations (PDEs) are classified as either elliptic, hyperbolic, or parabolic. Any second-order linear PDE in two variables can be written in the form :Au_ + 2Bu_ + Cu_ + Du_x + Eu_y + Fu +G= 0,\, whe ...
s. In the present context, the relevant elliptic equation is the condition for a function to be
harmonic A harmonic is a wave with a frequency that is a positive integer multiple of the ''fundamental frequency'', the frequency of the original periodic signal, such as a sinusoidal wave. The original signal is also called the ''1st harmonic'', t ...
relative to the Riemannian metric. The local solvability then states that any point has a neighborhood on which there is a harmonic function with nowhere-vanishing derivative. Isothermal coordinates are constructed from such a function in the following way. Harmonicity of is identical to the closedness of the differential 1-form \star du, defined using the
Hodge star operator In mathematics, the Hodge star operator or Hodge star is a linear map defined on the exterior algebra of a finite-dimensional oriented vector space endowed with a nondegenerate symmetric bilinear form. Applying the operator to an element of the ...
\star associated to the Riemannian metric. The Poincaré lemma thus implies the existence of a function on 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 and form a coordinate system on some neighborhood of . 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 In differential geometry, the Gaussian curvature or Gauss curvature of a surface at a point is the product of the principal curvatures, and , at the given point: K = \kappa_1 \kappa_2. The Gaussian radius of curvature is the reciprocal of . ...
takes the simpler form : K = -\frac e^ \left(\frac + \frac\right).


See also

*
Conformal map In mathematics, a conformal map is a function that locally preserves angles, but not necessarily lengths. More formally, let U and V be open subsets of \mathbb^n. A function f:U\to V is called conformal (or angle-preserving) at a point u_0\in ...
*
Liouville's equation :''For Liouville's equation in dynamical systems, see Liouville's theorem (Hamiltonian).'' : ''For Liouville's equation in quantum mechanics, see Von Neumann equation.'' : ''For Liouville's equation in Euclidean space, see Liouville–Bratu–Gelf ...
*
Quasiconformal map In mathematical complex analysis, a quasiconformal mapping, introduced by and named by , is a homeomorphism between plane domains which to first order takes small circles to small ellipses of bounded eccentricity. Intuitively, let ''f'' : ''D ...


Notes


References

* * * * * * . * * * * * Reprinted in: * * * * *


External links

* {{springer, title=Isothermal coordinates, id=p/i052890 Differential geometry Coordinate systems in differential geometry Partial differential equations