Understanding the Gauss Bonnet Theorem for 2D Riemann Surfaces

[lavinia]

I am looking at the proof of the Gauss Bonnet theorem for two dimensional Riemann surfaces.

There are many versions but this one escapes my intuition and I am asking for help "seeing" what it means though I can follow the arguments in the proofs.

View the surface as a Riemann surface and choose Isothermal coordinates around a point with scale factor L.

First how do I see that the Gauss curvature is -1/2L Laplacian(log L). Again I can do the computation.

I can see that if the surface is compact then L will be constant if it is harmonic and one would have a flat torus.

Now choose a meromorphic 1 form on the surface and write it as fdz in an isothermal coordinate chart around a single zero or pole of f.

Then the function L/|f|^2 extends to the whole surface and the exterior derivative of the complex differential of its logarithm equals -iKdS

How do I understand this function? For instance how is its complex differential related to the connection 1 form - if at all.

I apologize but I can not get the latex to work and then it has this big bug in it and I can't do anything. Help!

 

[Stephen Tashi]

I apologize but I can not get the latex to work and then it has this big bug in it and I can't do anything. Help!

To make LaTex work you have to keep refreshing the page. That is discussed in this thread: https://www.physicsforums.com/showthread.php?t=482096&highlight=LaTex