Residue calculus and gauss bonnet surfaces

[zwoodrow]

I am not a mathematician but I have noticed how strangley similar the treatments of curvature and residues are when you compare the residues of residue calculus and the curviture of the gauss bonet forumlation of surfaces. Is there some generalization of things that contains both of these formulations as a subset?

 

[zhentil]

You'll have to explain what you mean. On the surface, they don't have any relation at all.

 

[zwoodrow]

In both cases you are doing integration along a path that tells you something intrisic about the function/manifold inside of the curve as long as a certain point is inside the curve and not outside the curve. That simmalarity made me think that maybe the residue calculus was a subset of gauss bonnet.

 

[zhentil]

Ah, you're talking about Gauss-Bonnet for curves - then yes, the two are intimately related. On the Gauss Bonnet side, you're normalizing the curvature by integrating, and you're left with the winding number. On the Cauchy integral side, the integral of dz/z depends only on the winding number of the curve.

 

[lavinia]

Ah, you're talking about Gauss-Bonnet for curves - then yes, the two are intimately related. On the Gauss Bonnet side, you're normalizing the curvature by integrating, and you're left with the winding number. On the Cauchy integral side, the integral of dz/z depends only on the winding number of the curve.

when you say gauss bonnet for curves what do you mean?

 

[lavinia]

In a way the vector field proof of the Gauss Bonnet formula is a winding number proof.
If one normalizes a vector field with isolated zeros on an orientated surface to have length one away from its zeros, the the connection 1-form integrated over the image of a circle near a zero approximates the linking number of this image around the fiber circle above the zero. As one shrinks the circle towards the singularity this approximation improves and the linking number converges to a winding number.

i can elaborate this picture if you like.