View source for Conjugate isothermal coordinates

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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. [[#References|[a1]]]).

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  W. Blaschke,   K. Leichtweiss,   "Elementare Differentialgeometrie" , '''1''' , Springer  (1973)  pp. 160</TD></TR></table>