Difference between revisions of "Conjugate isothermal coordinates"
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Coordinates on a surface in which the second fundamental form is written as | 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 | 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==== | ====Comments==== | ||
Latest revision as of 17:46, 4 June 2020
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 |
Conjugate isothermal coordinates. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Conjugate_isothermal_coordinates&oldid=15512