# Why are they called isothermal coordinates?

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...)

• I keep trying to fix the display: the comma after the n=2 is too close. It looks fine in the preview. Any suggestions? Jul 16, 2010 at 13:47
• 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. Jul 16, 2010 at 14:00
• 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. May 11 at 17:34

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.