- Isothermal coordinates
In
mathematics , specifically indifferential geometry , isothermal coordinates on aRiemannian manifold are local coordinates where the metric isconformal to theEuclidean metric . This means that in isothermalcoordinates, theRiemannian 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 theCotton tensor .Isothermal coordinates on surfaces
The first result on the existence of isothermal coordinates was due to
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