Definition:Riemann Surface

Definition
A Riemann surface is a connected Hausdorff Space $$X$$ such that every point has a neighborhood homeomorphic to an open subset of the complex plane $$\C$$ and the transition maps are holomorphic.

More precisely, there exists a collection $$\mathcal U$$ of open sets $$U\subset X$$, each equipped with a function $$\phi_U: U \to \C$$, such that:
 * $$\bigcup \mathcal U = X$$; that is, $$\mathcal U$$ is an open cover of $$X$$.
 * Each $$\phi_U$$ is a homeomorphism between $$U$$ and an open subset of $$\C$$.
 * If $$U_1, U_2\in\mathcal U$$, then the function $$\phi_{U_2} \circ \phi_{U_1}^{-1}$$ (called the transition map) from $$U_1$$ to $$U_2$$ is holomorphic, where defined. (Note that the domain and range of the transition map are open subsets of $$\C$$.)

Properties
Every Riemann surface is path-connected, since it is connected and locally path-connected.

Every Riemann surface is second countable (that is, its topology has a countable base) by Radò's Theorem.

The universal cover of any Riemann Surface is conformally isomorphic to either the Riemann Sphere, the complex plane or the unit disk by the Riemann Uniformization Theorem.

It follows that every Riemann Surface admits a metric of constant curvature. In particular, every Riemann Surface is metrizable.

Elliptic, Parabolic and Hyperbolic Riemann Surface
A Riemann Surface $$X$$ is called
 * elliptic if its universal cover is the Riemann Sphere;
 * parabolic if its universal cover is the complex plane;
 * hyperbolic if its universal cover is the unit disk.

Equivalently, a surface is elliptic, parabolic or hyperbolic depending on whether it admits a metric of constant positive, zero or negative curvature, respectively.

The Riemann Sphere is the only elliptic Riemann Surface (up to conformal isomorphism).

A parabolic Riemann Surface is conformally isomorphic to either the complex plane, the punctured plane $$\C \setminus \left\{{0}\right\}$$, or a torus. Hence most Riemann Surfaces are hyperbolic.

Complex Manifold
A complex manifold of dimension $$n$$ is a $$2n \ $$-dimensional manifold where each point has an open neighborhood $$U$$ homeomorphic to a subset of $$\C^n$$ such that the transition maps are analytic.

In particular, a Riemann Surface is precisely a one-dimensional complex manifold.