Conjugate isothermal coordinates

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

$$ \tag{* } \textrm{ II } = - \Lambda ( u, v) ( du ^ {2} + dv ^ {2} ). $$

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

$$ \textrm{ II } = \Lambda ( u, v) ( du ^ {2} - dv ^ {2} ), $$

but in this case one often prefers the so-called asymptotic coordinates $ \widetilde{u} , \widetilde{v} $ for which

$$ \textrm{ II } = \widetilde \Lambda ( \widetilde{u} , \widetilde{v} ) \ d \widetilde{u} d \widetilde{v} . $$

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