Uniformization theorem
In mathematics, the uniformization theorem for surfaces says that any surface admits a metric of constant curvature in thermal coordinates. In other words, any surface has a complex structure and a metric of constant curvature - either 0, -1, or +1.
From this, a classification of surfaces follows. A surface is a quotient of one of: the complex plane (curvature 0), the Riemann sphere (curvature +1) or the unit disc (curvature -1 - hyperbolic plane) by a discrete group.
The first case is just a cylinder, torus or a complex plane.
In the second case we can have only the Riemann sphere itself.
The last case is the most important, and almost all surfaces are hyperbolic.
Referenced By
List of differential geometry topics | List of mathematical topics (S-U)
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