Isothermal coordinates
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In mathematics, specifically in differential geometry, isothermal coordinates on a Riemannian manifold are local coordinates where the metric is conformal to the Euclidean metric. This means that in isothermal coordinates, the metric tensor has the form
where φ is a smooth function. By a theorem of Hilbert, such coordinates always exist near any point when n=2, but when n > 2 a necessary and sufficient condition for their existence near every point is the vanishing of the the Weyl and Cotton curvature tensors.
Isothermal coordinates were first introduced by Gauss. In general, isothermal coordinates exist around any point on a two dimensional Riemannian manifold.
[edit] Properties
- In isothermal coordinates on a two dimensional manifold, the Gaussian curvature takes the simple form
[edit] See also
[edit] References
- Michael Spivak, A Comprehensive Introduction to Differential Geometry, 3rd ed., Publish or Perish Inc.
- Manfredo do Carmo, Differential Geometry of Curves and Surfaces, Prentice Hall.
