## 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 points $A$ and $B$ there exists an isometry of the surface that takes $A$ to $B$?

[Reposted from Math Stack Exchange.]

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I don't understand property 4. For instance, think of a square torus and take A=C. There's a whole circle of points of (small) constant distance from A, but the group of isometries of the torus that fix A is finite. – HW Nov 10 at 19:43
I suppose you are right. I will fix this. – Jonah Sinick Nov 10 at 20:18
Isn't property 4 false for all surfaces of genus $g \geq 2$? The isometry group is certainly contained in the automorphism group of the complex structure, which is finite by Hurwitz's theorem. – Will Sawin Nov 10 at 20:22
Probably what is meant by property 4 is that there is a local isometry which takes A to B. Perhaps one even wants to say that there is a local isometry from A to B inducing any given isometry of their tangent spaces. – unknown (google) Nov 10 at 20:43

I believe that on the first two questions, there is no answer (so they are not well-posed:-) The reason is that the flat structure on a torus and the moduli spaces of tori where very well understood long before the notions of "flat", "structure", "moduli space" and even "tori" were introduced. Mathematicians just did not think in these terms. Once these notions were introduced, the tori became a simple example.

Teichmuller was probably a person who made the notion of "moduli space" completely rigorous, though it was considered for the first time by Riemann. And Teichmuller considers tori at a great length to explain what he means. But certainly he did not find anything new about tori. This was just the simplest example. (Many people think, perhaps correctly, that Abel and Jacobi already knew everyhting about tori, though they did not use this word). Modular group was known to Gauss at about the same time. It is true, these things were developed for another 70 years or so, culminating in the Fricke-Klein and enormous French "treateases". But I know people who still prefer Jacobi's exposition, and even study Latin to read it:-)

The metric of constant negative curvature on surfaces of genus >1 is another story. This was introduced by Poincare. Even the Poincare metric on the unit disc was apparently Poincare's invention.
(Schwarz lemma was stated and proved much later, and not by Schwarz:-)

About the last question, I suppose Klein has to be credited. It was Klein who in his "Erlangen program" explained explicitly "what geometry is", and how it is connected with a group action of certain kind. Klein was in close collaboration with Lee at that time.

There is an excellent "primary source" on how all these things really developed: Klein's Lectures on the history of Math in XIX century.

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Dear Alexandre, Thank you for the thoughtful answer. – Jonah Sinick Nov 11 at 1:55
Poincare disc model (as well as Klein model) was invented by Beltrami. – Anton Petrunin Nov 11 at 4:10
Anton: that's interesting. Poincare does not mention Beltrami. Can you give a reference? – Alexandre Eremenko Nov 11 at 5:04
Poincare did mention Beltrami, he was only popularizing his construction. Eugenio Beltrami, Teoria fondamentale degli spazii di curvatura costante, Annali. di Mat., ser II 2 (1868), 232-255 – Anton Petrunin Nov 11 at 5:10
... which were both first invented by Bolyai in 1832. – Daniel Moskovich Nov 11 at 12:04