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1. Who first recognized that the torus supports a flat structure?

2. Who first characterized the moduli space of flat structures on the torus?

3. Who first recognized that the closed, orientable genus 2 supports a hyperbolic structure?

4. Who first thought of a geometrized surface in terms of the property that for any two pairs of points $(A, B)$ and $(C, D)$ such that the distance between $A$ and $B$ is equal to the distance between $C$ and $D$ there is exists an isometry of the surface that takes the pair $(A, B)$ A$to$(C, D)$?B$?

[Reposted from Math Stack Exchange.]

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# The history of the geometrization of closed surfaces

1. Who first recognized that the torus supports a flat structure?

2. Who first characterized the moduli space of flat structures on the torus?

3. Who first recognized that the closed, orientable genus 2 supports a hyperbolic structure?

4. Who first thought of a geometrized surface in terms of the property that for any two pairs of points $(A, B)$ and $(C, D)$ such that the distance between $A$ and $B$ is equal to the distance between $C$ and $D$ there is an isometry that takes the pair $(A, B)$ to $(C, D)$?

[Reposted from Math Stack Exchange.]