Riemann surface

Conversely, a two-dimensional real manifold can be turned into a Riemann surface (usually in several inequivalent ways) if and only if it is orientable and metrizable.

Given this, the sphere and torus admit complex structures but the Möbius strip, Klein bottle and real projective plane do not.

The two Riemann surfaces M and N are called biholomorphic (or conformally equivalent to emphasize the conformal point of view) if there exists a bijective holomorphic function from M to N whose inverse is also holomorphic (it turns out that the latter condition is automatic and can therefore be omitted).

Every non-compact Riemann surface admits non-constant holomorphic functions (with values in C).

In contrast, on a compact Riemann surface X every holomorphic function with values in C is constant due to the maximum principle.

This follows from the Kodaira embedding theorem and the fact there exists a positive line bundle on any complex curve.

If one global condition, namely compactness, is added, the surface is necessarily algebraic.

This feature of Riemann surfaces allows one to study them with either the means of analytic or algebraic geometry.

The corresponding statement for higher-dimensional objects is false, i.e. there are compact complex 2-manifolds which are not algebraic.

On the other hand, every projective complex manifold is necessarily algebraic, see Chow's theorem.

Geometrically, these correspond to surfaces with negative, vanishing or positive constant sectional curvature.

The description by the parameter τ gives the Teichmüller space of "marked" Riemann surfaces (in addition to the Riemann surface structure one adds the topological data of a "marking", which can be seen as a fixed homeomorphism to the torus).

However, in general the moduli space of Riemann surfaces of infinite topological type is too large to admit such a description.

These statements are clarified by considering the type of a Riemann sphere ^C with a number of punctures.

For example, hyperbolic Riemann surfaces are ramified covering spaces of the sphere (they have non-constant meromorphic functions), but the sphere does not cover or otherwise map to higher genus surfaces, except as a constant.