Difference between revisions of "Isothermal coordinates"
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| − | <table><TR><TD valign="top">[a1]</TD> <TD valign="top"> W. Blaschke, K. Leichtweiss, "Elementare Differentialgeometrie" , Springer (1973)</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> |
Revision as of 12:11, 27 September 2012
Coordinates of a two-dimensional Riemannian space in which the square of the line element has the form:
 |
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:
 |
where 
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:
 |
Here, frequent use is made of coordinates 
which are naturally connected with isothermal coordinates and in which the square of the line element has the form:
 |
In this case the lines 
and 
are isotropic geodesics and the coordinate system 
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 |
Isothermal coordinates. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Isothermal_coordinates&oldid=13178