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"?
Best Answer
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.