Is Horn-torus a valid genus-1 Riemann Surface?

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SUMMARY

The horn torus is not a valid genus-1 Riemann surface due to its failure to meet the criteria of a manifold. Specifically, the central point of the horn torus resembles two cones touching at their tips, which disqualifies it from being a Riemann surface. A Riemann surface must be a 2-dimensional topological manifold, and the horn torus does not satisfy this requirement. Therefore, it is essential to understand the properties of manifolds to accurately classify surfaces like the horn torus.

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jackmell
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The plot below is a horn torus. Is that a valid genus-1 normal Riemann surface? I believe it is but I'm just a novice. I'm unsure about the single point in the center and if it "technically" still has a hole in it. Maybe I need to review that.
 

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A Riemann surface is, in particular, a 2 dimensional topological manifold. At the single point at the center, this space looks like 2 cones with their tips touching. Because of that, The horned torus is not a manifold and hence not a Riemann surface.
 
Ok, thanks.
 

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