Meromorphic Functions on Riemann surfaces

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lavinia
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In Felix's Klein's pamphlet, "On Riemann's Theory of Algebraic Functions and their Integrals" he describes ways to construct divergence free irrotational flows on a compact Riemann surfaces such as the torus.

One method is simply to cover the surface with a conducting material and place two oppositely charged electric poles - using a battery - at two points on the surface. The potential difference at these poles will create an electric field - which may be viewed as a flow.

The field will have two oppositely oriented logarithmic singularities at the poles and will be regular (finite) everywhere else.

My question is: How does one know that the electric field will not circulate around one of the handles of the surface? For instance on the torus, how does one know that it will not circulate around the meridian? By this I mean, how does one know that the integral of the field along a closed homotopically non-trivial loop will be zero? Or is this not true?

If it is true then one can coalesce the two logarithmic singularities to get a new field whose potential is a meromorphic function. But if the field circulates around one of the non-trivial loops its potential will be multi-valued.
 
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