Conjugate isothermal coordinates

Coordinates on a surface in which the second fundamental form is written as

(*)

Conjugate isothermal coordinates can always be introduced in a sufficiently small neighbourhood of an elliptic point of a regular surface. In a sufficiently small neighbourhood of a hyperbolic point of a regular surface one can introduce coordinates in which

but in this case one often prefers the so-called asymptotic coordinates for which

Comments

This notion is rarely used in Western literature. As the second fundamental form changes its sign if the orientation of the surface path is reversed, the minus in (*) is not important and is, in fact, commonly deleted.

Conjugate isothermal coordinates are also called affine isothermal coordinates (cf. [a1]).

References

[a1]	W. Blaschke, K. Leichtweiss, "Elementare Differentialgeometrie" , 1 , Springer (1973) pp. 160