Editing Torus (section)

=== Conformal classification of flat tori ===

In the study of [[Riemann surface|Riemann surfaces]], one says that any two smooth compact geometric surfaces are "conformally equivalent" when there exists a smooth homeomorphism between them that is both angle-preserving and orientation-preserving. The [[Uniformization theorem]] guarantees that every Riemann surface is [[Conformal map|conformally equivalent]] to one that has constant [[Gaussian curvature]]. In the case of a torus, the constant curvature must be zero. Then one defines the "[[moduli space]]" of the torus to contain one point for each conformal equivalence class, with the appropriate topology. It turns out that this moduli space ''M'' may be identified with a punctured sphere that is smooth except for two points that have less angle than 2π (radians) around them: One has total angle π and the other has total angle 2π/3. 

''M'' may be turned into a compact space ''M*'' – topologically equivalent to a sphere – by adding one additional point that represents the limiting case as a rectangular torus approaches an aspect ratio of 0 in the limit. The result is that this compactified moduli space  is a sphere with ''three'' points each having less than 2π total angle around them. (Such a point is termed a "cusp", and may be thought of as the vertex of a cone, also called a "conepoint".) This third conepoint will have zero total angle around it. Due to symmetry, ''M*'' may be constructed by glueing together two congruent [[geodesic triangle]]s in the [[hyperbolic plane]] along their (identical) boundaries, where each triangle has angles of {{math|π/2}}, {{math|π/3}}, and {{math|0}}. (The three angles of a hyperbolic triangle T determine T up to congruence.) As a result, the [[Gauss–Bonnet theorem]] shows that the area of each triangle can be calculated as {{math|1=π − (π/2 + π/3 + 0) = π/6}}, so it follows that the compactified moduli space ''M*'' has area equal to {{math|π/3}}.

The other two cusps occur at the points corresponding in ''M*'' to (a) the square torus (total angle {{math|π}}) and (b) the hexagonal torus (total angle {{math|2π/3}}). These are the only conformal equivalence classes of flat tori that have any conformal automorphisms other than those generated by translations and negation.