In mathematics, particularly in complex analysis, a **Riemann surface** is a one-dimensional complex manifold. These surfaces were first studied by and are named after Bernhard Riemann. 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 several sheets glued together.

- Definitions
- Examples
- Further definitions and properties
- Orientability
- Functions
- Analytic vs. algebraic
- Classification of Riemann surfaces
- Elliptic Riemann surfaces
- Parabolic Riemann surfaces
- Hyperbolic Riemann surfaces
- Maps between Riemann surfaces
- Punctured spheres
- Ramified covering spaces
- Isometries of Riemann surfaces
- Function-theoretic classification
- See also
- Theorems regarding Riemann surfaces
- Notes
- References
- External links

The main interest in 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 and metrizable. So the sphere and torus admit complex structures, but the Möbius strip, Klein bottle and real 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 connected complex manifold of complex dimension one. This means that*X*is a connected Hausdorff space that is endowed with an atlas of charts to the open unit disk of the complex plane: for every point*x*∈*X*there is a neighbourhood of*x*that is homeomorphic to the open unit disk of the complex plane, and the transition maps between two overlapping charts are required to be holomorphic. - A Riemann surface is an oriented manifold of (real) dimension two – a two-sided surface – together with a conformal structure. Again, manifold means that locally at any point
*x*of*X*, the space is homeomorphic to a subset of 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.^{ [1] }

- 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 non-empty open subset of the complex plane can be viewed as a Riemann surface in a natural way. More generally, every non-empty 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*),

*P*is a complex polynomial of degree 2*g*+ 1. - All compact Riemann surfaces are algebraic curves since they can be embedded into some . This follows from the Kodaira embedding theorem and the fact there exists a positive line bundle on any complex curve.
^{ [2] } - Important examples of non-compact Riemann surfaces are provided by analytic continuation.

*f*(*z*) = arcsin*z**f*(*z*) = log*z**f*(*z*) =*z*^{1/2}*f*(*z*) =*z*^{1/3}*f*(*z*) =*z*^{1/4}

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* ∘ *f* ∘ *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.

Each Riemann surface, being a complex manifold, is orientable as a real manifold. For complex charts *f* and *g* with transition function *h* = *f*(*g*^{−1}(*z*)), *h* can be considered 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.

Every non-compact Riemann surface admits non-constant holomorphic functions (with values in **C**). In fact, every non-compact Riemann surface is a Stein manifold.

In contrast, on a compact Riemann surface *X* every holomorphic function with values in **C** is constant due to the maximum principle. However, there always exist 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 Siegel (1955). Meromorphic functions can be given fairly explicitly, in terms of Riemann theta functions and the Abel–Jacobi map of the surface.

The existence of non-constant 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 3-space. 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 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.

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 j-invariant *j*(*E*), which can be used to determine *τ* and hence a torus.

The set of all Riemann surfaces can be divided into three subsets: hyperbolic, parabolic and elliptic Riemann surfaces. Geometrically, these correspond to surfaces with negative, vanishing or positive constant sectional curvature. That is, every connected Riemann surface admits a unique complete 2-dimensional real Riemann metric with constant curvature equal to or which belongs to the conformal class of Riemannian metrics determined by its structure as a Riemann surface. This can be seen as a consequence of the existence of isothermal coordinates.

In complex analytic terms, the Poincaré–Koebe uniformization theorem (a generalization of the Riemann mapping theorem) states that every simply connected Riemann surface is conformally equivalent to one of the following:

- The Riemann sphere , which is isomorphic to the ;
- The complex plane ;
- The open disk which is isomorphic to the upper half-plane .

A Riemann surface is elliptic, parabolic or hyperbolic according to whether its universal cover is isomorphic to , or . The elements in each class admit a more precise description.

The Riemann sphere is the only example, as there is no group acting on it by biholomorphic transformations freely and properly discontinuously and so any Riemann surface whose universal cover is isomorphic to must itself be isomorphic to it.

If is a Riemann surface whose universal cover is isomorphic to the complex plane then it is isomorphic one of the following surfaces:

- itself;
- The quotient ;
- A quotient where with .

Topologically there are only three types: the plane, the cylinder and the torus. But while in the two former case the (parabolic) Riemann surface structure is unique, varying the parameter in the third case gives non-isomorphic Riemann surfaces. The description by the parameter 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). To obtain the analytic moduli space (forgetting the marking) one takes the quotient of Teichmüller space by the mapping class group. In this case it is the modular curve.

In the remaining cases is a hyperbolic Riemann surface, that is isomorphic to a quotient of the upper half-plane by a Fuchsian group (this is sometimes called a Fuchsian model for the surface). The topological type of can be any orientable surface save the torus and sphere.

A case of particular interest is when is compact. Then its topological type is described by its genus . Its Teichmüller space and moduli space are -dimensional. A similar classification of Riemann surfaces of finite type (that is homeomorphic to a closed surface minus a finite number of points) can be given. However in general the moduli space of Riemann surfaces of infinite topological type is too large to admit such a description.

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 holomorphic 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)!

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.

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 non-constant 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 non-constant meromorphic functions), but the sphere does not cover or otherwise map to higher genus surfaces, except as a constant.

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 leaving invariant the set containing only infinity and zero: 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.
- 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
*g*≥ 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 is known that every finite group can be realized as the full group of isometries of some Riemann surface.
^{ [3] }- 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 the unique simple group of order 168, which is the second-smallest non-abelian simple group. This group is isomorphic to both PSL(2,7) and PSL(3,2).
- 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 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 non-constant negative subharmonic functions on the surface and is otherwise called *hyperbolic*.^{ [4] }^{ [5] } 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.

- ↑ See (Jost 2006 , Ch. 3.11) for the construction of a corresponding complex structure.
- ↑ Nollet, Scott. "KODAIRA'S THEOREM AND COMPACTIFICATION OF MUMFORD'S MODULI SPACE Mg" (PDF).
- ↑ Greenberg, L. (1974). "Maximal groups and signatures".
*Discontinuous Groups and Riemann Surfaces: Proceedings of the 1973 Conference at the University of Maryland*. Ann. Math. Studies.**79**. pp. 207–226. ISBN 0691081387. - ↑ 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, ISBN 9781468480382

In the mathematical field of complex analysis, a **meromorphic function** on an open subset *D* of the complex plane is a function that is holomorphic on all of *D**except* for a set of isolated points, which are poles of the function. The term comes from the Ancient Greek *meros* (μέρος), meaning "part".

In complex analysis, a pole is a certain type of singularity of a function, nearby which the function behaves relatively regularly, in contrast to essential singularities, such as 0 for the logarithm function, and branch points, such as 0 for the complex square root function.

In mathematics, **complex geometry** is the study of complex manifolds, complex algebraic varieties, and functions of several complex variables. Application of transcendental methods to algebraic geometry falls in this category, together with more geometric aspects of complex analysis.

The **Riemann–Roch theorem** is an important theorem in mathematics, specifically in complex analysis and algebraic geometry, for the computation of the dimension of the space of meromorphic functions with prescribed zeros and allowed poles. It relates the complex analysis of a connected compact Riemann surface with the surface's purely topological genus *g*, in a way that can be carried over into purely algebraic settings.

In mathematics, a **modular form** is a (complex) analytic function on the upper half-plane satisfying a certain kind of functional equation with respect to the group action of the modular group, and also satisfying a growth condition. The theory of modular forms therefore belongs to complex analysis but the main importance of the theory has traditionally been in its connections with number theory. Modular forms appear in other areas, such as algebraic topology, sphere packing, and string theory.

In mathematics, the **uniformization theorem** says that every simply connected Riemann surface is conformally equivalent to one of three Riemann surfaces: the open unit disk, the complex plane, or the Riemann sphere. In particular it implies that every Riemann surface admits a Riemannian metric of constant curvature. For compact Riemann surfaces, those with universal cover the unit disk are precisely the hyperbolic surfaces of genus greater than 1, all with non-abelian fundamental group; those with universal cover the complex plane are the Riemann surfaces of genus 1, namely the complex tori or elliptic curves with fundamental group **Z**^{2}; and those with universal cover the Riemann sphere are those of genus zero, namely the Riemann sphere itself, with trivial fundamental group.

The theory of **functions of several complex variables** is the branch of mathematics dealing with complex-valued functions. The name of the field dealing with the properties of **function of several complex variables** is called **several complex variables**, that has become a common name for that whole field of study and Mathematics Subject Classification has, as a top-level heading. A function is n-tuples of complex numbers, classically studied on the complex coordinate space .

In the mathematical field of complex analysis, **Nevanlinna theory** is part of the theory of meromorphic functions. It was devised in 1925, by Rolf Nevanlinna. Hermann Weyl has called it "one of the few great mathematical events of century." The theory describes the asymptotic distribution of solutions of the equation *f*(*z*) = *a*, as *a* varies. A fundamental tool is the Nevanlinna characteristic *T*(*r*, *f*) which measures the rate of growth of a meromorphic function.

In mathematics, **Hodge theory**, named after W. V. D. Hodge, is a method for studying the cohomology groups of a smooth manifold *M* using partial differential equations. The key observation is that, given a Riemannian metric on *M*, every cohomology class has a canonical representative, a differential form that vanishes under the Laplacian operator of the metric. Such forms are called **harmonic**.

In differential geometry and complex geometry, a **complex manifold** is a manifold with an atlas of charts to the open unit disk in **C**^{n}, such that the transition maps are holomorphic.

In mathematics, **algebraic geometry and analytic geometry** are two closely related subjects. While algebraic geometry studies algebraic varieties, analytic geometry deals with complex manifolds and the more general analytic spaces defined locally by the vanishing of analytic functions of several complex variables. The deep relation between these subjects has numerous applications in which algebraic techniques are applied to analytic spaces and analytic techniques to algebraic varieties.

In number theory and algebraic geometry, a **modular curve***Y*(Γ) is a Riemann surface, or the corresponding algebraic curve, constructed as a quotient of the complex upper half-plane **H** by the action of a congruence subgroup Γ of the modular group of integral 2×2 matrices SL(2, **Z**). The term modular curve can also be used to refer to the **compactified modular curves***X*(Γ) which are compactifications obtained by adding finitely many points to this quotient. The points of a modular curve parametrize isomorphism classes of elliptic curves, together with some additional structure depending on the group Γ. This interpretation allows one to give a purely algebraic definition of modular curves, without reference to complex numbers, and, moreover, prove that modular curves are defined either over the field of rational numbers **Q** or a cyclotomic field **Q**(ζ_{n}). The latter fact and its generalizations are of fundamental importance in number theory.

In contexts including complex manifolds and algebraic geometry, a **logarithmic** differential form is a meromorphic differential form with poles of a certain kind. The concept was introduced by Deligne.

In mathematics, **Hurwitz's automorphisms theorem** bounds the order of the group of automorphisms, via orientation-preserving conformal mappings, of a compact Riemann surface of genus *g* > 1, stating that the number of such automorphisms cannot exceed 84(*g* − 1). A group for which the maximum is achieved is called a **Hurwitz group**, and the corresponding Riemann surface a **Hurwitz surface**. Because compact Riemann surfaces are synonymous with non-singular complex projective algebraic curves, a Hurwitz surface can also be called a **Hurwitz curve**. The theorem is named after Adolf Hurwitz, who proved it in.

In mathematics, a **Fuchsian model** is a representation of a hyperbolic Riemann surface *R* as a quotient of the upper half-plane **H** by a Fuchsian group. Every hyperbolic Riemann surface admits such a representation. The concept is named after Lazarus Fuchs.

In mathematics, with special application to complex analysis, a *normal family* is a pre-compact subset of the space of continuous functions. Informally, this means that the functions in the family are not widely spread out, but rather stick together in a somewhat "clustered" manner. Sometimes, if each function in a normal family *F* satisfies a particular property , then the property also holds for each limit point of the set *F*.

In mathematics, the **Teichmüller space** of a (real) topological surface , is a space that parametrizes complex structures on up to the action of homeomorphisms that are isotopic to the identity homeomorphism. Teichmüller spaces are named after Oswald Teichmüller.

In mathematics, a **Klein surface** is a dianalytic manifold of complex dimension 1. Klein surfaces may have a boundary and need not be orientable. Klein surfaces generalize Riemann surfaces. While the latter are used to study algebraic curves over the complex numbers analytically, the former are used to study algebraic curves over the real numbers analytically. Klein surfaces were introduced by Felix Klein in 1882.

In mathematics, the **Riemann sphere**, named after Bernhard Riemann, is a model of the **extended complex plane**, the complex plane plus a point at infinity. This extended plane represents the **extended complex numbers**, that is, the complex numbers plus a value ∞ for infinity. With the Riemann model, the point "∞" is near to very large numbers, just as the point "0" is near to very small numbers.

In mathematics and especially complex geometry, the **Kobayashi metric** is a pseudometric intrinsically associated to any complex manifold. It was introduced by Shoshichi Kobayashi in 1967. **Kobayashi hyperbolic** manifolds are an important class of complex manifolds, defined by the property that the Kobayashi pseudometric is a metric. Kobayashi hyperbolicity of a complex manifold *X* implies that every holomorphic map from the complex line **C** to *X* is constant.

- Farkas, Hershel M.; Kra, Irwin (1980),
*Riemann Surfaces*(2nd ed.), Berlin, New York: Springer-Verlag, ISBN 978-0-387-90465-8 - Pablo Arés Gastesi,
*Riemann Surfaces Book*. - Hartshorne, Robin (1977),
*Algebraic Geometry*, Berlin, New York: Springer-Verlag, ISBN 978-0-387-90244-9, MR 0463157, OCLC 13348052 , esp. chapter IV. - Jost, Jürgen (2006),
*Compact Riemann Surfaces*, Berlin, New York: Springer-Verlag, pp. 208–219, ISBN 978-3-540-33065-3 - 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 978-3-03719-029-6, MR 2284826, S2CID 119593165 - Lawton, Sean; Peterson, Elisha (2009), Papadopoulos, Athanase (ed.),
*Handbook of Teichmüller theory. Vol. II*, IRMA Lectures in Mathematics and Theoretical Physics,**13**, European Mathematical Society (EMS), Zürich, arXiv: math/0511271 , doi:10.4171/055, ISBN 978-3-03719-055-5, MR 2524085, S2CID 16687772 - Papadopoulos, Athanase, ed. (2012),
*Handbook of Teichmüller theory. Vol. III*, IRMA Lectures in Mathematics and Theoretical Physics,**19**, European Mathematical Society (EMS), Zürich, doi:10.4171/103, ISBN 978-3-03719-103-3 - Siegel, Carl Ludwig (1955), "Meromorphe Funktionen auf kompakten analytischen Mannigfaltigkeiten",
*Nachrichten der Akademie der Wissenschaften in Göttingen. II. Mathematisch-Physikalische Klasse*,**1955**: 71–77, ISSN 0065-5295, MR 0074061 - Weyl, Hermann (2009) [1913],
*The concept of a Riemann surface*(3rd ed.), New York: Dover Publications, ISBN 978-0-486-47004-7, MR 0069903

- "Riemann surface",
*Encyclopedia of Mathematics*, EMS Press, 2001 [1994]

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