Riemann surface

For the Riemann surface of a subring of a field, see Zariski–Riemann space.
In mathematics, particularly in complex analysis, a Riemann surface, first studied by and named after Bernhard Riemann, is a onedimensional 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 multivalued functions such as the square root and other algebraic functions, or the logarithm.
Every Riemann surface is a twodimensional 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 twodimensional real manifold can be turned into a Riemann surface (usually in several inequivalent ways) if and only if it is orientable and metrizable. 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.
Contents
Definitions
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 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.
 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 socalled 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.^{[1]}
Further information: complex manifold and conformal geometryExamples
 The complex plane C is 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 nonsingular. For example, the torus C/(Z + τ Z), where τ is a complex nonreal 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 noncompact Riemann surfaces are provided by analytic continuation.
Further definitions and properties
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.
Orientability
We noted in the preamble that all Riemann surfaces, like all complex manifolds, are orientable as a real manifold. The reason is that for complex charts f and g with transition function h = f(g^{−1}(z)) we can consider h as a map from an open set of R^{2} to R^{2} whose Jacobian in a point z is just the real linear map given by multiplication by the complex number h'(z). However, the real determinant of multiplication by a complex number α equals α^{2}, so the Jacobian of h has positive determinant. Consequently the complex atlas is an oriented atlas.
Functions
Every noncompact Riemann surface admits nonconstant holomorphic functions (with values in C). In fact, every noncompact Riemann surface is a Stein manifold.
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 nonconstant 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 Siegel (1955).
Analytic vs. algebraic
The above fact about existence of nonconstant meromorphic functions can be used to show that any compact Riemann surface is a projective variety, i.e. can be given by polynomial equations inside a projective space. Actually, it can be shown that every compact Riemann surface can be embedded into complex projective 3space. This is a surprising theorem: Riemann surfaces are given by locally patching charts. If one global condition, namely compactness, is added, the surface is necessarily algebraic. This feature of Riemann surfaces allows to study them with either the means of analytic or algebraic geometry. The corresponding statement for higherdimensional objects is false, i.e. there are compact complex 2manifolds which are not algebraic. On the other hand, every projective complex manifold is necessarily algebraic, see Chow's theorem.
As an example, consider the torus T := C/(Z + τ Z). The Weierstrass function belonging to the lattice Z + τ Z is a meromorphic function on T. This function and its derivative generate the function field of T. There is an equation
where the coefficients g_{2} and g_{3} depend on τ, thus giving an elliptic curve E_{τ} in the sense of algebraic geometry. Reversing this is accomplished by the jinvariant j(E), which can be used to determine τ and hence a torus.
Classification of Riemann surfaces
The realm of Riemann surfaces can be divided into three regimes: hyperbolic, parabolic and elliptic Riemann surfaces, with the distinction given by the uniformization theorem. Geometrically, these correspond to negative curvature, zero curvature/flat, and positive curvature: stating the uniformization theorem in terms of conformal geometry, every connected Riemann surface X admits a unique complete 2dimensional real Riemann metric with constant curvature −1, 0 or 1 inducing the same conformal structure – every metric is conformally equivalent to a constant curvature metric. The surface X is called hyperbolic, parabolic, and elliptic, respectively.
For simply connected Riemann surfaces, the uniformization theorem states that every simply connected Riemann surface is conformally equivalent to one of the following:
 elliptic
 the Riemann sphere C ∪ {∞}, also denoted P^{1}C
 parabolic
 the complex plane C, or
 hyperbolic
 the open disk D := {z ∈ C : z < 1} or equivalently the upper halfplane H := {z ∈ C : Im(z) > 0}.
The existence of these three types parallels the several nonEuclidean geometries.
The general technique of associating to 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.
Elliptic Riemann surfaces
By definition, these are the surfaces X with constant curvature +1. The Riemann sphere C ∪ {∞} is the only example. (Elliptic functions are examples of parabolic Riemann surfaces. The naming comes from the history: elliptic functions are associated to elliptic integrals, which in turn show up in calculating the circumference of ellipses).
Parabolic Riemann surfaces
By definition, these are the surfaces X with constant curvature 0. Equivalently, by the uniformization theorem, the universal cover of X has to be the complex plane.
There are then three possibilities for X. It can be the plane itself, an annulus, or a torus
 T := C / (Z ⊕ τZ).
The set of representatives of the cosets are called fundamental domains.
Care must be taken insofar as two tori are always homeomorphic, but in general not biholomorphic to each other. This is the first appearance of the problem of moduli. The modulus of a torus can be captured by a single complex number τ with positive imaginary part. In fact, the marked moduli space (Teichmüller space) of the torus is biholomorphic to the upper halfplane or equivalently the open unit disk.
Hyperbolic Riemann surfaces
The Riemann surfaces with curvature −1 are called hyperbolic. This group is the "biggest" one.
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 Poincarémetric 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 hyperelliptic 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 halfplane 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 isomorphic to a compact surface with a finite number of points removed. 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 GaussBonnet theorem to the area of the fundamental polygon.
Maps between Riemann surfaces
The geometric classification is reflected in maps between Riemann surfaces, as detailed in Liouville's theorem and the Little Picard theorem: maps from hyperbolic to parabolic to elliptic are easy, but maps from elliptic to parabolic or parabolic to hyperbolic are very constrained (indeed, generally constant!). There are inclusions of the disc in the plane in the sphere: but any meromorphic map from the sphere to the plane is constant, any holomorphic map from the plane into the unit disk is constant (Liouville's theorem), and in fact any holomorphic map from the plane into the plane minus two points is constant (Little Picard theorem)!
Punctured spheres
These statements are clarified by considering the type of a Riemann sphere with a number of punctures. With no punctures, it is the Riemann sphere, which is elliptic. With one puncture, which can be placed at infinity, it is the complex plane, which is parabolic. With two punctures, it is the punctured plane or alternatively annulus or cylinder, which is parabolic. With three or more punctures, it is hyperbolic – compare pair of pants. One can map from one puncture to two, via the exponential map (which is entire and has an essential singularity at infinity, so not defined at infinity, and misses zero and infinity), but all maps from zero punctures to one or more, or one or two punctures to three or more are constant.
Ramified covering spaces
Continuing in this vein, compact Riemann surfaces can map to surfaces of lower genus, but not to higher genus, except as constant maps. This is because holomorphic and meromorphic maps behave locally like so nonconstant maps are ramified covering maps, and for compact Riemann surfaces these are constrained by the Riemann–Hurwitz formula in algebraic topology, which relates the Euler characteristic of a space and a ramified cover.
For example, hyperbolic Riemann surfaces are ramified covering spaces of the sphere (they have nonconstant meromorphic functions), but the sphere does not cover or otherwise map to higher genus surfaces, except as a constant.
Isometries of Riemann surfaces
The isometry group of a uniformized Riemann surface (equivalently, the conformal automorphism group) reflects its geometry:
 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 halfplane is the real Möbius group; this is conjugate to the automorphism group of the disk.
 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°.
 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.^{[2]}
 For genus 2 the order is maximized by the Bolza surface, with order 48.
 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 secondsmallest nonabelian simple group.
 For genus 4, Bring's surface is a highly symmetric surface.
 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 fourthsmallest nonabelian simple group.
Functiontheoretic classification
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.^{[3]}^{[4]} 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 functiontheoretic classification. For example, the Riemann surface consisting of "all complex numbers but 0 and 1" is parabolic in the functiontheoretic classification but it is hyperbolic in the geometric classification.
See also
 Dessin d'enfant
 Kähler manifold
 Lorentz surface
 Theorems regarding Riemann surfaces
 Mapping class group
 branching theorem
 Hurwitz's automorphisms theorem
 identity theorem for Riemann surfaces
 Riemann–Roch theorem
 Riemann–Hurwitz formula
Notes
 ^ See (Jost 2006, Ch. 3.11) for the construction of a corresponding complex structure.
 ^ L.Greenberg, Maximal groups and signatures, Ann. Math. Studies 79 (1974) 207226
 ^ Ahlfors, Lars; Sario, Leo (1960), Riemann Surfaces (1st ed.), Princeton, New Jersey: Princeton University Press, p. 204
 ^ Rodin, Burton; Sario, Leo (1968), Principal Functions (1st ed.), Princeton, New Jersey: D. Von Nostrand Company, Inc., p. 199
References
 Farkas, Hershel M.; Kra, Irwin (1980), Riemann Surfaces (2nd ed.), Berlin, New York: SpringerVerlag, ISBN 9780387904658
 Pablo Arés Gastesi, Riemann Surfaces Book.
 Hartshorne, Robin (1977), Algebraic Geometry, Berlin, New York: SpringerVerlag, ISBN 9780387902449, OCLC 13348052, MR0463157, esp. chapter IV.
 Jost, Jürgen (2006), Compact Riemann Surfaces, Berlin, New York: SpringerVerlag, pp. 208–219, ISBN 9783540330653
 Papadopoulos, Athanase, ed. (2007), Handbook of Teichmüller theory. Vol. I, IRMA Lectures in Mathematics and Theoretical Physics, 11, European Mathematical Society (EMS), Zürich, doi:10.4171/029, ISBN 9783037190296, MR2284826
 Papadopoulos, Athanase, ed. (2009), Handbook of Teichmüller theory. Vol. II, IRMA Lectures in Mathematics and Theoretical Physics, 13, European Mathematical Society (EMS), Zürich, doi:10.4171/055, ISBN 9783037190555, MR2524085
 Siegel, Carl Ludwig (1955), "Meromorphe Funktionen auf kompakten analytischen Mannigfaltigkeiten", Nachrichten der Akademie der Wissenschaften in Göttingen. II. MathematischPhysikalische Klasse 1955: 71–77, ISSN 00655295, MR0074061
 Weyl, Hermann (2009) [1913], The concept of a Riemann surface (3rd ed.), New York: Dover Publications, ISBN 9780486470047, MR0069903, http://www.archive.org/details/dieideederrieman00weyluoft
External links
Categories: Complex analysis
 Riemann surfaces
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