Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

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

    lavinia

    User Avatar
    Science Advisor
    Gold Member

    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

    lavinia

    User Avatar
    Science Advisor
    Gold Member

    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
    thanks
     
  6. May 28, 2010 #5
    Eh? What do you mean by "all surfaces"?
     
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook




Similar Discussions: Conformally Flat and Einstein Geometry
  1. Conformal equivalence (Replies: 9)

Loading...