# Conformally Flat and Einstein Geometry

1. May 26, 2010

### charlynd

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. May 27, 2010

### lavinia

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.

$$ds^2 = Edu^2 + 2Fdudv + Gdv^2$$

Once this Beltrami equation is derived one then appeals to the existence of solutions to the PDE to get the existence of the coordinates.

3. May 27, 2010

### lavinia

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.

4. May 27, 2010

### charlynd

thanks

5. May 28, 2010

### shoehorn

Eh? What do you mean by "all surfaces"?