Conformally Flat and Einstein Geometry

[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?

 

[lavinia]

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?

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

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.

 

[lavinia]

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?

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.

 

[charlynd]

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

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.

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.

thanks

 

[shoehorn]

All surfaces are conformally flat.

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