TITLE: Who first discovered that the torus supports a flat structure?
QUESTION [8 upvotes]: Who first recognized that there exists a homogenous metric on the closed genus 1 orientable surface?

REPLY [6 votes]: I believe from this paper that Clifford was the one of the first people to describe a flat metric on the torus; he was the first to explicitly embed the torus into $\mathbb{R}^4$ in the standard flat way; this paper also states that he was one of the first to speak of the torus as a parallelogram with opposite edges identified. It's a fun paper to read!