I read this Wikipedia article just today. There, I first learned more details about the complex manifold called the Riemann surface:
In mathematics, particularly in complex analysis, a Riemann surface, first studied by and named after Bernhard Riemann, is a one-dimensional complex manifold. Riemann surfaces can be thought of as "deformed versions" of the complex plane: locally near every point they look like patches of the complex plane, but the global topology can be quite different. For example, they can look like a sphere or a torus or a couple of sheets glued together.
The main point of Riemann surfaces is that holomorphic functions may be defined between them. Riemann surfaces are nowadays considered the natural setting for studying the global behavior of these functions, especially multi-valued functions such as the square root and other algebraic functions, or the logarithm.
Every Riemann surface is a two-dimensional real analytic manifold (i.e., a surface), but it contains more structure (specifically a complex structure) which is needed for the unambiguous definition of holomorphic functions. A two-dimensional real manifold can be turned into a Riemann surface (usually in several inequivalent ways) if and only if it is orientable. So the sphere and torus admit complex structures, but the Möbius strip, Klein bottle and projective plane do not.
Geometrical facts about Riemann surfaces are as "nice" as possible, and they often provide the intuition and motivation for generalizations to other curves, manifolds or varieties. The Riemann–Roch theorem is a prime example of this influence.
There are several equivalent definitions of a Riemann surface:
1. A Riemann surface X is a complex manifold of complex dimension one. This means that X is a Hausdorff topological space endowed with an atlas: for every point x ∈ X there is a neighbourhood containing x homeomorphic to the unit disk of the complex plane. The map carrying the structure of the complex plane to the Riemann surface is called a chart. Additionally, the transition maps between two overlapping charts are required to be holomorphic.
2. A Riemann surface is a manifold of (real) dimension two – a surface – together with a conformal structure. Again, manifold means that locally at any point x of X, the space is supposed to be like the real plane. The supplement "Riemann" signifies that X is endowed with an additional structure which allows angle measurement on the manifold, namely an equivalence class of so-called Riemannian metrics. Two such metrics are considered equivalent if the angles they measure are the same. Choosing an equivalence class of metrics on X is the additional datum of the conformal structure.
A complex structure gives rise to a conformal structure by choosing the standard Euclidean metric given on the complex plane and transporting it to X by means of the charts. Showing that a conformal structure determines a complex structure is more difficult.
Isometries of Riemann surfaces
The isometry group of a uniformized Riemann surface (equivalently, the conformal automorphism group)
reflects its geometry:
1. Genus 0
The isometry group of the sphere is the Möbius group of projective transforms of the complex line. The isometry group of the plane is the subgroup fixing infinity, and of the punctured plane is the subgroup fixing infinity and zero as a set: either fixing them both, or interchanging them (1/z).
The isometry group of the upper half-plane is the real Möbius group; this is conjugate to the automorphism group of the disk.
2. Genus 1
The isometry group of a torus is in general translations (as an Abelian variety), though the square lattice and hexagonal lattice have addition symmetries from rotation by 90° and 60°.
3. For genus ≥ 2
The isometry group is finite, and has order at most 84(g − 1), by Hurwitz's automorphisms theorem; surfaces that realize this bound are called Hurwitz surfaces.
It's known that every finite group can be realized as the full group of isometries of some Riemann surface.
Riemann Surfaces with Maximal Orders
A. For genus 2 the order is maximized by the Bolza surface, with order 48.
B. For genus 3 the order is maximized by the Klein quartic, with order 168; this is the first Hurwitz surface, and its automorphism group is isomorphic to PSL(2,7), the second-smallest non-abelian simple group.
C. For genus 4, Bring's surface is a highly symmetric surface.
D. For genus 7 the order is maximized by the Macbeath surface, with order 504; this is the second Hurwitz surface, and its automorphism group is isomorphic to PSL(2,8), the fourth-smallest non-abelian simple group.
The classification scheme above is typically used by geometers. There is a different classification for Riemann surfaces which is typically used by complex analysts. It employs a different definition for "parabolic" and "hyperbolic". In this alternative classification scheme, a Riemann surface is called parabolic if there are no nonconstant negative subharmonic functions on the surface and is otherwise called hyperbolic. This class of hyperbolic surfaces is further subdivided into subclasses according to whether function spaces other than the negative subharmonic functions are degenerate, e.g. Riemann surfaces on which all bounded holomorphic functions are constant, or on which all bounded harmonic functions are constant, or on which all positive harmonic functions are constant, etc.
To avoid confusion, call the classification based on metrics of constant curvature the geometric classification, and the one based on degeneracy of function spaces the function-theoretic classification. For example, the Riemann surface consisting of "all complex numbers but 0 and 1" is parabolic in the function-theoretic classification but it is hyperbolic in the geometric classification.