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

jackmell

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.

quasar987

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.

jackmell

Ok, thanks.

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jackmell	Is Horn-torus a valid genus-1 Riemann Surface? Tweet 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. Attached Thumbnails

quasar987 Recognitions: Gold Member Homework Help Science Advisor	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.