Isothermal coordinates

In mathematics, specifically in differential geometry, isothermal coordinates on a Riemannian manifold are local coordinates where the metric is
conformal to the Euclidean metric. This means that in isothermalcoordinates, the Riemannian metric locally has the form: $g = e^varphi (dx_1^2 + cdots + dx_n^2),$ where $varphi$ is a smooth function.

Isothermal coordinates on surfaces were first introduced by Gauss. Korn and Lichtenstein proved that isothermal coordinates exist around any point on a two dimensional Riemannian manifold. On higher dimensional Riemannian manifolds a necessary and sufficient condition for their local existence is the vanishing of the Weyl tensor and the Cotton tensor.

Isothermal coordinates on surfaces

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Isothermal coordinates

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