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Who first recognized that there exists a homogenous metric on the closed genus 1 orientable surface?

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IDK. May be whoever had a sheet of paper, and noticed that you can roll it to form a tube without stretching. And then observed that if the material does stretch a bit, you could bend it to form a torus? Or may be a map-maker during a coffee break, who noticed that longitudes and latitudes would work very well on the surface of a donut. I really think that this metric on the torus is so natural that nobody would want his/her name associated with its discovery - and/or it was independently arrived at by very many people. – Jyrki Lahtonen Nov 10 '12 at 7:21

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