Isothermal coordinates

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In mathematics, specifically in differential geometry, an isothermal coordinate system on a Riemannian manifold M is a particular choice of local coordinates on some open neighbourhood of M in which the metric on M is `similar' to a Euclidean metric.


[edit] Definition

Suppose M is a smooth manifold with a Riemannian metric g. Also, suppose \phi: U \subset M \to \mathbb{R}^n is a coordinate patch on some open set in M; this gives us a frame of tangent vectors e_i = \phi^*(\partial/\partial x_i) on U, and a dual coframe εi such that εi(ej) = δij.

Then we say that φ is an isothermal coordinate system on U if g has the form g = \lambda (\sum_i \epsilon_i \otimes \epsilon_i) where λ is some smooth function on U with λ > 0.

[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.
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