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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
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025050/c0250501.png" /></td> <td valign="top" style="width:5%;text-align:right;">(*)</td></tr></table>
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$$ \tag{* }
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\textrm{ II }  = - \Lambda ( u, v) ( du  ^ {2} + dv  ^ {2} ).
 +
$$
  
 
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
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025050/c0250502.png" /></td> </tr></table>
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$$
 
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\textrm{ II }  = \Lambda ( u, v) ( du  ^ {2} - dv  ^ {2} ),
but in this case one often prefers the so-called asymptotic coordinates <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025050/c0250503.png" /> for which
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$$
 
 
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025050/c0250504.png" /></td> </tr></table>
 
  
 +
but in this case one often prefers the so-called asymptotic coordinates  $  \widetilde{u}  , \widetilde{v}  $
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for which
  
 +
$$
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\textrm{ II }  =  \widetilde \Lambda  ( \widetilde{u}  , \widetilde{v}  ) \
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d \widetilde{u}  d \widetilde{v}  .
 +
$$
  
 
====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
How to Cite This Entry:
Conjugate isothermal coordinates. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Conjugate_isothermal_coordinates&oldid=46472
This article was adapted from an original article by D.D. Sokolov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article