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On a Riemannian manifold, a coordinate system is called "isothermal" if the Riemannian metric in those coordinates is conformal to the Euclidean metric:

$$g_{ij} = e^{f} \delta_{ij}$$

My question is: Why are such coordinate systems called "isothermal"? It must have something to do with classical thermal physics. I tried looking for a reason online, with no success.

It is well known that when the dimension $n=2$, there always exist isothermal coordinates, and this is probably where they were first introduced. So maybe the nomenclature has something to do with heat diffusion in the plane?

(The reason I ask is because I am planning to give a seminar talk next week giving a proof that such coordinates exist when $n=2$, and thought it would be nice to explain to the students where the name comes from...)

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  • $\begingroup$ I keep trying to fix the display: the comma after the n=2 is too close. It looks fine in the preview. Any suggestions? $\endgroup$
    – Spiro Karigiannis
    Commented Jul 16, 2010 at 13:47
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    $\begingroup$ As the display depends highly on which browser you are using and which computer you are using and which set of fonts your are using, I think trying to prettify the display is a futile exercise. Just make sure it is syntactically correct. $\endgroup$
    – Willie Wong
    Commented Jul 16, 2010 at 14:00
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    $\begingroup$ A Riemannian manifold has an intrinsic Laplacian, called the Laplace-Beltrami operator. Isothermal coordinates are harmonic in the sense that they belong to the kernel of this operator. On an oriented Riemannian 2-manifold, every harmonic function u has a harmonic conjugate function v, defined by the relation dv = * du, where * denotes the Hodge star operator. Near a point where du is non-zero, the pair (u,v) then provides an isothermal coordinate system, and every isothermal coordinate system arises this way. $\endgroup$
    – Claude LeBrun
    Commented May 11, 2022 at 17:34

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Isothermal coordinates are harmonic. In other words, it solves $\triangle_g u = 0$. So locally it is a stationary solution of the heat equation. In physics, for a steady state distribution of temperatures, each level set is called an isotherm.

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  • $\begingroup$ Thanks! I should have thought of that, since I knew that isothermal coordinates are harmonic. In general, though, one can consider "harmonic coordinates" which are not necessarily isothermal. So it may not be the best nomenclature, but I guess it's here to stay. $\endgroup$
    – Spiro Karigiannis
    Commented Jul 16, 2010 at 14:50
  • $\begingroup$ Well, in two dimensions where the original studies were done, there's no obstruction to conformally flat, so harmonic and isothermal coordinates are more or less the same (one also needs the two coordinates to be harmonic conjugates of each other). Also now isothermal coordinates also picked up the additional meaning to be coordinates in which it is conformally Euclidean. So it does mean something more specific than harmonic. $\endgroup$
    – Willie Wong
    Commented Jul 16, 2010 at 15:14
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According to Gray, Abbena and Salamon, that's the name given to such coordinate systems by Gabriel Lamé in his 1833 study of heat transfer. The reason is, if you've got a thermally isolated surface of constant heat conduction, the constant coordinate lines are isotherms iff the coordinates are isothermal.

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