Isothermal coordinates (or conformal coordinates)

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

\begin{matrix} (1) & g_{i j} = e^{ρ} I d = Λ^{2} I d \end{matrix}

The coefficient $Λ^{2}$ 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 $Δ_{g} u = 0$ . 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

Gaussian curvature

In isothermal coordinates, Gaussian curvature takes the simpler form (wikipedia):

K = - \frac{1}{2} e^{- ρ} (\frac{\partial^{2} ρ}{\partial u^{2}} + \frac{\partial^{2} ρ}{\partial v^{2}}) = - \frac{Δ log (Λ)}{Λ^{2}}

where $Λ^{2} = e^{ρ}$ and $Δ$ is the Laplacian operator. Here $Λ$ 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 $δ$ in the chart $(u, v)$ , in the surface we will have moved a quantity $Λ δ$ (@needham2021visual page 41).