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Conformally Flat and Einstein Geometry

  1. May 26, 2010 #1
    Could somebody explain me what conformally flat is?

    How to prove a 2D geometry as conformally flat, for example:
    ds^2 = \phi(dx^2-dy^2) ; phi(x,y)

    What is class of Einstein Geometry?

    How to classify Einstein Geometry in 2D?
  2. jcsd
  3. May 27, 2010 #2


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    All surfaces are conformally flat. The proof is to show that around any point there are Isothermal coordinates. These are conformal maps from a domain in the complex plain onto a neighborhood on the surface.

    Existence of Isothermal coordinates is a classical result and involves solving a PDE known as the Beltrami equation. It is not difficult to derive. You should try it.

    Start with arbitrary coordinates

    [tex] ds^2 = Edu^2 + 2Fdudv + Gdv^2[/tex]

    Once this Beltrami equation is derived one then appeals to the existence of solutions to the PDE to get the existence of the coordinates.
  4. May 27, 2010 #3


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    A manifold with constant sectional curvature is Einstein. For a surface this is a manifold of constant Gauss curvature. That any riemann surface has a metric of constant Gauss curvature is another classical theorem.
  5. May 27, 2010 #4
  6. May 28, 2010 #5
    Eh? What do you mean by "all surfaces"?
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