On a pseudo-Riemannian manifold, a coordinate system is called "isothermal" if the Riemannian metric in those coordinates is conformal to the Euclidean metric:
$$ g_{ij} = e^{\rho} Id=\Lambda^2 Id \tag{1} $$The coefficient $\Lambda^2$ is called conformal factor. It is a measure of how distorted is the surface in every point with respect tot a plane.
Example: see stereographic projection#First approach.
It is well known that when the dimension $n=2$, there always exist isothermal coordinates, and this is probably where they were first introduced.
They solve $\Delta_g u=0$. 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.
See this Q&A
In isothermal coordinates, Gaussian curvature takes the simpler form (wikipedia):
$$ K=-\frac{1}{2} e^{-\rho}\left(\frac{\partial^{2} \rho}{\partial u^{2}}+\frac{\partial^{2} \rho}{\partial v^{2}}\right)=-\frac{\Delta \mbox{log}(\Lambda)}{\Lambda^2} $$where $\Lambda^2=e^{\rho}$ and $\Delta$ is the Laplacian operator. Here $\Lambda$ is the expansion factor of a vector in the surface with respect to itself expressed in the chart. That is, if we move a quantity $\delta$ in the chart $(u,v)$, in the surface we will have moved a quantity $\Lambda\delta$ (@needham2021visual page 41).
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Author of the notes: Antonio J. Pan-Collantes
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