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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.

- 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 an 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.
- A Riemann surface is a Riemannian manifold of (real) dimension two - hence the name Riemann 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 a so-called Riemannian metric g, which allows angle measurement on the manifold. Two such metrics are considered equivalent if the angles they measure are the same. Choosing a metric, and hence an equivalence class of metrices 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.

- The complex plane C is perhaps the most basic Riemann surface. The map f(z) = z (the identity map) defines a chart for C, and {f} is an atlas for C. The map g(z) = z
^{*}(the conjugate map) also defines a chart on C and {g} is an atlas for C. The charts f and g are not compatible, so this endows C with two distinct Riemann surface structures. In fact, given a Riemann surface X and its atlas A, the conjugate atlas B = {f^{*}: f ∈ A} is never compatible with A, and endows X with a distinct, incompatible Riemann structure. - In an analogous fashion, every open subset of the complex plane can be viewed as a Riemann surface in a natural way. More generally, every open subset of a Riemann surface is a Riemann surface.
- Let S = C ∪ {∞} and let f(z) = z where z is in S {∞} and g(z) = 1 / z where z is in S {0} and 1/∞ is defined to be 0. Then f and g are charts, they are compatible, and { f, g } is an atlas for S, making S into a Riemann surface. This particular surface is called the Riemann sphere because it can be interpreted as wrapping the complex plane around the sphere. Unlike the complex plane, it is compact.
- The theory of compact Riemann surfaces can be shown to be equivalent to that of projective algebraic curves that are defined over the complex numbers and non-singular. For example, the torus C/(Z + τ Z), where τ is a complex non-real number, corresponds, via the Weierstrass elliptic function associated to the lattice Z + τ Z, to an elliptic curve given by an equation

- y
^{2}= x^{3}+ a x + b.

- Tori are the only Riemann surfaces of genus one, surfaces of higher genera g are provided by the hyperelliptic surfaces

- y
^{2}= P(x),

- where P is a complex polynomial of degree 2g + 1.

- Important examples of non-compact Riemann surfaces are provided by analytic continuation.

As with any map between complex manifolds, a function f: M → N between two Riemann surfaces M and N is called holomorphic if for every chart g in the atlas of M and every chart h in the atlas of N, the map h o f o g^{-1} is holomorphic (as a function from C to C) wherever it is defined. The composition of two holomorphic maps is holomorphic. 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). Two conformally equivalent Riemann surfaces are for all practical purposes identical.

In contrast, on a compact Riemann surface X every holomorphic function with value in C is constant due to the maximum principle. However, there always exists non-constant meromorphic functions (holomorphic functions with values in the Riemann sphere C ∪ {∞}). More precisely, the function field of X is a finite extension of C(t), the function field in one variable, i.e. any two meromorphic functions are algebraically dependent. This statement generalizes to higher dimensions, see .

As an example, consider the torus T := C/(Z+τ Z). The Weierstrass function $wp\_tau(z)$ belonging to the lattice Z+τ Z is a meromorphic function on T. This function and its derivative $wp\text{'}\_tau(z)$ generate the function field of T. There is an equation

- $$

- the complex plane C
- the Riemann sphere C ∪ {∞}, also denoted P
^{1}C

or

- the open disk D := {z ∈ C : |z| < 1} or equivalently the upper half-plane H := {z ∈ C : Im(z) > 0}.

According to the equivalence of two definitions given above, the uniformization theorem can also be stated in terms of conformal geometry: every connected Riemann surface X admits a unique complete 2-dimensional real Riemann metric with constant curvature −1, 0 or 1 inducing the same conformal structure. The surface X is called hyperbolic, parabolic, and elliptic, respectively. The existence of these three types parallels the several (non-)Euclidean geometries.

The general technique of associating a manifold X its universal cover Y, and expressing the original X as the quotient of Y by the group of deck transformations gives a first overview over Riemann surfaces.

There are then three possibilities for X. It can be the plane itself, an annulus, or a torus

- T := C / (Z ⊕ τZ).

The celebrated Riemann mapping theorem states that any simply connected strict subset of the complex plane is biholomorphic to the unit disk. Therefore the open disk with the Poincaré-metric of constant curvature −1 is the local model of any hyperbolic Riemann surface. According to the uniformization theorem above, all hyperbolic surfaces are quotients of the unit disk.

Examples include all surfaces with genus g > 1 such as hyper-elliptic curves.

For every hyperbolic Riemann surface, the fundamental group is isomorphic to a Fuchsian group, and thus the surface can be modelled by a Fuchsian model H/Γ where H is the upper half-plane and Γ is the Fuchsian group. The set of representatives of the cosets of H/Γ are free regular sets and can be fashioned into metric fundamental polygons. Quotient structures as H/Γ are generalized to Shimura varieties.

Unlike elliptic and parabolic surfaces, no classification of the hyperbolic surfaces is possible. Any connected open strict subset of the plane gives a hyperbolic surface; consider the plane minus a Cantor set. A classification is possible for surfaces of finite type: those with finitely generated fundamental group. Any one of these has a finite number of moduli and so a finite dimensional Teichmüller space. The problem of moduli (solved by Lars Ahlfors and extended by Lipman Bers) was to justify Riemann's claim that for a closed surface of genus g, 3g − 3 complex parameters suffice.

When a hyperbolic surface is compact, then the total area of the surface is 4π(g − 1), where g is the genus of the surface; the area is obtained by applying the Gauss-Bonnet theorem to the area of the fundamental polygon.

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 of nonzero complex numbers is parabolic in the function-theoretic classification but it is hyperbolic in the geometric classification.

- One of M. C. Escher's works, Print Gallery, is laid out on a cyclically growing grid that has been described as a Riemann surface.
- In Aldous Huxley's novel Brave New World, "Riemann Surface Tennis" is a popular game.

- Kähler manifold
- Theorems regarding Riemann surfaces

- Pablo Arés Gastesi, Riemann Surfaces Book.
- , esp. chapter IV.

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Last updated on Tuesday September 23, 2008 at 09:45:24 PDT (GMT -0700)

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Last updated on Tuesday September 23, 2008 at 09:45:24 PDT (GMT -0700)

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