TITLE: The history of the geometrization of closed surfaces
QUESTION [9 upvotes]: Who first recognized that the torus supports a flat structure? 
Who first characterized the moduli space of flat structures on the torus?
Who first recognized that the closed, orientable genus 2 supports a hyperbolic structure?
Who first thought of a geometrized surface in terms of the property that for any two points $A$ and $B$ there exists neighborhoods $U$ and $V$ of $A$ and $B$, respectively, and an isometry from $U$ to $V$?

[Reposted from Math Stack Exchange.]

REPLY [2 votes]: See this Wikipedia article, and the references therein. Jeremy Gray's article is particularly enlightening.