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If I understand correctly, Gauss proved that given any oriented Riemannian surface, one can find a complex structure on the surface so that the metric on the charts is just $f|dz|$, where $f>0$.

I've heard these coordinates referred as "conformal coordinates," which makes sense, but I've also heard of them referred to as "isothermal coordinates."

Why are these coordinates called "isothermal"? Who named it "isothermal"? Does it have anything to do with physics, or is it kind of like "Inertia"?

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    $\begingroup$ "Why are they called... " $\endgroup$ – Mariano Suárez-Álvarez Jan 22 '11 at 18:48
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    $\begingroup$ This might have something to do with the fact that the real and imaginary parts (which one can think of as coordinates) of a holomorphic function are harmonic, and therefore are steady-state solutions of the heat equation. $\endgroup$ – Rahul Jan 22 '11 at 18:57
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    $\begingroup$ A question with the exact same title has been asked on MathOverflow. To which I gave pretty much the same answer given by Mariano below. $\endgroup$ – Willie Wong Jan 23 '11 at 0:11
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If $(u,v)$ are isothermal coordinates, then $u$ and $v$ are harmonic functions with respect to the Laplace-Beltrami operator on your Riemannian manifold, that is $\Delta u=\Delta v=0$.

Now, the equation $\Delta f=0$ characterizes the stationary states for the heat equation. The level curves for a harmonic function are therefore the isothermal curves for some heat distribution.

It follows that if $(u,v)$ is an isothermal system of coordinates, then the level curves, i.e., the coordinate lines, are isothermal curves.

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It is similar to isotherms(countours that connect equal points of temperature). See the wikipedia article on isothermal coordinates.

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    $\begingroup$ I don't think I understand... How do I get these contours, and what is the quantity that is analogous to "temperature" in this setting? $\endgroup$ – Braindead Jan 22 '11 at 18:44
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I can recommend you the book "Modern Differential Geometry of Curves and Surfaces with Mathematica" Alfred Gray. This would be useful.

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    $\begingroup$ You're recommending a reading an entire book to answer the question of why a concept has a particular name? Or is there a specific page you have in mind? $\endgroup$ – Zev Chonoles Aug 6 '13 at 18:28
  • $\begingroup$ I agree with Zev Chonoles, is there a particular passage of that book that you had in mind that answers the question? $\endgroup$ – Ataraxia Aug 6 '13 at 18:46

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