Residue calculus and gauss bonnet surfaces

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Discussion Overview

The discussion revolves around the relationship between residue calculus and the Gauss-Bonnet theorem as applied to surfaces and curves. Participants explore potential connections and generalizations between the concepts of curvature in the Gauss-Bonnet formulation and residues in complex analysis.

Discussion Character

  • Exploratory
  • Technical explanation
  • Debate/contested

Main Points Raised

  • One participant notes a similarity between the treatments of curvature in the Gauss-Bonnet formulation and residues in residue calculus, suggesting a possible generalization that includes both.
  • Another participant challenges this view, stating that there appears to be no relation between the two concepts on the surface.
  • A participant elaborates that both concepts involve integration along a path that reveals intrinsic properties about the function or manifold, provided a certain point is inside the curve.
  • One participant affirms the connection by referencing Gauss-Bonnet for curves, indicating that the two concepts are indeed related through the normalization of curvature and the winding number.
  • A later reply discusses the vector field proof of the Gauss-Bonnet formula as analogous to a winding number proof, suggesting that the linking number converges to a winding number as the circle around a singularity shrinks.

Areas of Agreement / Disagreement

Participants express differing views on the relationship between residue calculus and the Gauss-Bonnet theorem. While some see a connection, others argue against it, indicating that the discussion remains unresolved.

Contextual Notes

Participants have not fully defined the terms and concepts being discussed, which may lead to varying interpretations. The relationship between the two mathematical frameworks is not clearly established and relies on specific conditions and assumptions.

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

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