Isothermal coordinates

Coordinates of a two-dimensional Riemannian space in which the square of the line element has the form:

$$ds ^ {2} = \lambda ( \xi , \eta ) ( d \xi ^ {2} + d \eta ^ {2} ).$$

Isothermal coordinates specify a conformal mapping of the two-dimensional Riemannian manifold into the Euclidean plane. Isothermal coordinates can always be introduced in a compact domain of a regular two-dimensional manifold. The Gaussian curvature can be calculated in isothermal coordinates by the formula:

$$k = - \frac{\Delta \mathop{\rm ln} \lambda } \lambda ,$$

where $\Delta$ is the Laplace operator.

Isothermal coordinates are also considered in two-dimensional pseudo-Riemannian spaces; the square of the line element then has the form:

$$ds ^ {2} = \psi ( \xi , \eta ) ( d \xi ^ {2} - d \eta ^ {2} ).$$

Here, frequent use is made of coordinates $\mu , \nu$ which are naturally connected with isothermal coordinates and in which the square of the line element has the form:

$$ds ^ {2} = \lambda ( \mu , \nu ) d \mu d \nu .$$

In this case the lines $\mu = \textrm{ const }$ and $\nu = \textrm{ const }$ are isotropic geodesics and the coordinate system $\mu , \nu$ is called isotropic. Isotropic coordinates are extensively used in general relativity theory.

References

[a1]	W. Blaschke, K. Leichtweiss, "Elementare Differentialgeometrie" , Springer (1973) MR0350630 Zbl 0264.53001