Uniformization Theorem - Geometric Classification of Surfaces

Geometric Classification of Surfaces

On an oriented surface, a Riemannian metric naturally induces an almost complex structure as follows: For a tangent vector v we define J(v) as the vector of the same length which is orthogonal to v and such that (v, J(v)) is positively oriented. On surfaces any almost complex structure is integrable, so this turns the given surface into a Riemann surface.

From this, a classification of metrizable surfaces follows. A connected metrizable surface is a quotient of one of the following by a free action of a discrete subgroup of an isometry group:

1. the sphere (curvature +1)
2. the Euclidean plane (curvature 0)
3. the hyperbolic plane (curvature −1).

The first case includes all surfaces with positive Euler characteristic: the sphere and the real projective plane. The second includes all surfaces with vanishing Euler characteristic: the Euclidean plane, cylinder, Möbius strip, torus, and Klein bottle. The third case covers all surfaces with negative Euler characteristic: almost all surfaces are hyperbolic. For closed surfaces, this classification is consistent with the Gauss–Bonnet theorem, which implies that for a closed surface with constant curvature, the sign of that curvature must match the sign of the Euler characteristic.

The positive/flat/negative classification corresponds in algebraic geometry to Kodaira dimension −1,0,1 of the corresponding complex algebraic curve.

For Riemann surfaces, Rado's theorem implies that the surface is automatically second countable. For general surfaces this is no longer true, so for the classification above one needs to assume that the surface is second countable (or metrizable). The Prüfer surface is an example of a surface with no (Riemannian) metric.