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Coordinates of a two-dimensional Riemannian space in which the square of the line element has the form:
 
Coordinates of a two-dimensional Riemannian space in which the square of the line element has the form:
  
<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/i/i052/i052890/i0528901.png" /></td> </tr></table>
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$$
 +
ds  ^ {2}  = \lambda ( \xi , \eta ) ( d \xi  ^ {2} + d \eta  ^ {2} ).
 +
$$
  
 
Isothermal coordinates specify a conformal mapping of the two-dimensional Riemannian manifold into the Euclidean plane. Isothermal coordinates can always be introduced in a compact domain of a regular two-dimensional manifold. The Gaussian curvature can be calculated in isothermal coordinates by the formula:
 
Isothermal coordinates specify a conformal mapping of the two-dimensional Riemannian manifold into the Euclidean plane. Isothermal coordinates can always be introduced in a compact domain of a regular two-dimensional manifold. The Gaussian curvature can be calculated in isothermal coordinates by the formula:
  
<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/i/i052/i052890/i0528902.png" /></td> </tr></table>
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$$
 +
= -  
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\frac{\Delta  \mathop{\rm ln}  \lambda } \lambda
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,
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$$
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052890/i0528903.png" /> is the [[Laplace operator|Laplace operator]].
+
where $  \Delta $
 +
is the [[Laplace operator|Laplace operator]].
  
 
Isothermal coordinates are also considered in two-dimensional pseudo-Riemannian spaces; the square of the line element then has the form:
 
Isothermal coordinates are also considered in two-dimensional pseudo-Riemannian spaces; the square of the line element then has the form:
  
<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/i/i052/i052890/i0528904.png" /></td> </tr></table>
+
$$
 +
ds  ^ {2}  = \psi ( \xi , \eta ) ( d \xi  ^ {2} - d \eta  ^ {2} ).
 +
$$
  
Here, frequent use is made of coordinates <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052890/i0528905.png" /> which are naturally connected with isothermal coordinates and in which the square of the line element has the form:
+
Here, frequent use is made of coordinates $  \mu , \nu $
 
+
which are naturally connected with isothermal coordinates and in which the square of the line element has the form:
<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/i/i052/i052890/i0528906.png" /></td> </tr></table>
 
 
 
In this case the lines <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052890/i0528907.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052890/i0528908.png" /> are isotropic geodesics and the coordinate system <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052890/i0528909.png" /> is called isotropic. Isotropic coordinates are extensively used in general relativity theory.
 
  
 +
$$
 +
ds  ^ {2}  =  \lambda ( \mu , \nu )  d \mu  d \nu .
 +
$$
  
 +
In this case the lines  $  \mu = \textrm{ const } $
 +
and  $  \nu = \textrm{ const } $
 +
are isotropic geodesics and the coordinate system  $  \mu , \nu $
 +
is called isotropic. Isotropic coordinates are extensively used in general relativity theory.
  
 
====Comments====
 
====Comments====
 
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  W. Blaschke,  K. Leichtweiss,  "Elementare Differentialgeometrie" , Springer  (1973)  {{MR|0350630}} {{ZBL|0264.53001}} </TD></TR></table>
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  W. Blaschke,  K. Leichtweiss,  "Elementare Differentialgeometrie" , Springer  (1973)  {{MR|0350630}} {{ZBL|0264.53001}} </TD></TR></table>

Latest revision as of 22:13, 5 June 2020


Coordinates of a two-dimensional Riemannian space in which the square of the line element has the form:

$$ ds ^ {2} = \lambda ( \xi , \eta ) ( d \xi ^ {2} + d \eta ^ {2} ). $$

Isothermal coordinates specify a conformal mapping of the two-dimensional Riemannian manifold into the Euclidean plane. Isothermal coordinates can always be introduced in a compact domain of a regular two-dimensional manifold. The Gaussian curvature can be calculated in isothermal coordinates by the formula:

$$ k = - \frac{\Delta \mathop{\rm ln} \lambda } \lambda , $$

where $ \Delta $ is the Laplace operator.

Isothermal coordinates are also considered in two-dimensional pseudo-Riemannian spaces; the square of the line element then has the form:

$$ ds ^ {2} = \psi ( \xi , \eta ) ( d \xi ^ {2} - d \eta ^ {2} ). $$

Here, frequent use is made of coordinates $ \mu , \nu $ which are naturally connected with isothermal coordinates and in which the square of the line element has the form:

$$ ds ^ {2} = \lambda ( \mu , \nu ) d \mu d \nu . $$

In this case the lines $ \mu = \textrm{ const } $ and $ \nu = \textrm{ const } $ are isotropic geodesics and the coordinate system $ \mu , \nu $ is called isotropic. Isotropic coordinates are extensively used in general relativity theory.

Comments

References

[a1] W. Blaschke, K. Leichtweiss, "Elementare Differentialgeometrie" , Springer (1973) MR0350630 Zbl 0264.53001
How to Cite This Entry:
Isothermal coordinates. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Isothermal_coordinates&oldid=28225
This article was adapted from an original article by D.D. Sokolov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article