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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.in summary,The two formulations of curvature and residue calculus are related in that on the Gauss Bonnet side you normalize the curv

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

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zhentil said:

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

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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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