Riemann surfaces

Classification of Riemann surfaces

The set of all Riemann surfaces can be divided into three subsets: hyperbolic, parabolic and elliptic Riemann surfaces. Geometrically, these correspond to surfaces with negative, vanishing or positive constant sectional curvature. That is, every connected Riemann surface ${\displaystyle X}$ admits a unique complete 2-dimensional real Riemann metric with constant curvature equal to ${\displaystyle -1}$, ${\displaystyle 0}$ or ${\displaystyle 1}$ that belongs to the conformal class of Riemannian metrics determined by its structure as a Riemann surface. This can be seen as a consequence of the existence of isothermal coordinates.