# Riemann surface

A Riemann surface is a connected one-dimensional complex manifold: a surface that looks locally like an open set in the complex plane, with coordinate changes that are holomorphic. It is the natural setting in which multi-valued analytic functions such as the square root and the logarithm become single-valued, and it is the object on which the theory of analytic functions of one complex variable is built.<sup>[1](https://www.math.lmu.de/~forster/lehre/vorlB6s_rie.html)</sup>

| Key fact | Detail |
|---|---|
| Definition | A smooth surface with coordinate charts to \( \mathbb{C} \) whose overlap maps satisfy the Cauchy-Riemann equations, equivalently a complex 1-dimensional manifold<sup>[2](https://www.math.uci.edu/~jviaclov/courses/218BC_2022.pdf)</sup> |
| Origin | Introduced in Bernhard Riemann's Göttingen doctoral thesis, examined 16 December 1851<sup>[3](https://mathshistory.st-andrews.ac.uk/Biographies/Riemann/)</sup> |
| Original purpose | Domains on which "multi-valued" functions like the square root or logarithm become single-valued, as surfaces covering the complex plane<sup>[1](https://www.math.lmu.de/~forster/lehre/vorlB6s_rie.html)</sup> |
| Uniformization | Every simply connected Riemann surface is isomorphic to exactly one of the Riemann sphere, the complex plane, or the unit disk<sup>[4](https://www.mat.univie.ac.at/~armin/lect/Riemann_surfaces.pdf)</sup> |
| Algebraicity | Every compact Riemann surface is isomorphic to a smooth algebraic curve over \( \mathbb{C} \)<sup>[5](https://math.uchicago.edu/~may/REU2015/REUPapers/Smith.pdf)</sup> |
| Compactness | A compact Riemann surface carries no holomorphic functions besides the constants; meromorphic functions and forms carry the theory<sup>[6](https://www.mathematik.uni-muenchen.de/~wehler/Lehrveranstaltungen_WS_2019_2020.php)</sup> |
| Canonical divisor | On a compact Riemann surface of genus \( g \), every canonical divisor \( K \) has degree \( 2g-2 \)<sup>[7](https://people.math.osu.edu/cueto.5/teaching/8140/Sp23/index.html)</sup> |

## Definition

A Riemann surface is a smooth two-dimensional surface \( M \) together with a collection of coordinate charts \( (U_\alpha, \phi_\alpha) \) covering it, where each \( \phi_\alpha \) maps \( U_\alpha \) into \( \mathbb{C} \), and the transition maps \( \phi_\alpha \circ \phi_\beta^{-1} \) between overlapping charts satisfy the Cauchy-Riemann equations, which is exactly the condition that they be holomorphic.<sup>[2](https://www.math.uci.edu/~jviaclov/courses/218BC_2022.pdf)</sup> In the language of manifolds, a Riemann surface is a one-dimensional complex manifold, locally like an open set in the complex plane.<sup>[1](https://www.math.lmu.de/~forster/lehre/vorlB6s_rie.html)</sup> A map \( f: M \to N \) between two such surfaces is called holomorphic if it is holomorphic with respect to the local charts.<sup>[8](https://www.sciencedirect.com/topics/mathematics/riemann-surface)</sup>

The one-complex-dimensional condition has a topological consequence: every Riemann surface is canonically orientable, and a compact one is classified up to homeomorphism by a single integer, its genus \( g \), the number of handles.<sup>[9](https://arxiv.org/abs/2410.13273)</sup> Topologically, a compact Riemann surface of genus \( g \) is a sphere with \( g \) handles.<sup>[10](https://arxiv.org/abs/0903.2156)</sup>

## Origin in Riemann's dissertation

The concept goes back to [Bernhard Riemann](https://www.edgepedia.org/bernhard-riemann)'s doctoral thesis *Grundlagen für eine allgemeine Theorie der Functionen einer veränderlichen complexen Grösse* ([Göttingen](https://www.edgepedia.org/gottingen), 1851), which was examined on 16 December 1851.<sup>[3](https://mathshistory.st-andrews.ac.uk/Biographies/Riemann/)</sup><sup> • </sup><sup>[11](https://www.maths.tcd.ie/pub/HistMath/People/Riemann/Grund/)</sup> Since that dissertation, Riemann surfaces have for the most part been considered as suitable domains of definition for analytic functions.<sup>[12](https://www.cambridge.org/core/journals/mathematical-proceedings-of-the-cambridge-philosophical-society/article/riemann-surfaces-of-field-extensions/60DF036951C4FBF4DA0121FFFCBB7AF2)</sup> Riemann demonstrated, first in an 1851 work and then in a paper from 1857 that reached a broader audience, that these surfaces admit a classification by a number, subsequently named the genus, which equals the largest possible count of closed curves that can be traced on the surface while it remains in one piece.<sup>[13](https://mathshistory.st-andrews.ac.uk/SH/riemann_sh.pdf)</sup>

The original motivation was the treatment of multi-valued functions. The square root and the logarithm cannot be defined as single-valued holomorphic functions on the whole complex plane, but they can be treated in a satisfactory way using Riemann surfaces covering the plane.<sup>[1](https://www.math.lmu.de/~forster/lehre/vorlB6s_rie.html)</sup> In the early history of the subject, Riemann surfaces were understood as branched coverings of \( \mathbb{C} \) or of the sphere, on which they serve as the domains where multi-valued holomorphic or meromorphic functions are naturally defined.<sup>[8](https://www.sciencedirect.com/topics/mathematics/riemann-surface)</sup> For a multi-valued analytic function \( w = f(z) \), its Riemann surface \( R_f \) provides a geometric means of uniformization: the multi-valued relation is replaced by two single-valued relations \( w = F(p) \) and \( z = g(p) \) for \( p \in R_f \), giving a single-valued expression over the whole domain of the complete analytic function.<sup>[14](https://encyclopediaofmath.org/wiki/Riemann_surface)</sup>

## Holomorphic maps and branched covers

Holomorphic maps between Riemann surfaces are the morphisms of the theory, and their local behavior is rigid. A non-constant holomorphic map \( F: R \to S \) makes \( R \) a branched cover of \( S \): the set \( B \subset S \) of branch points is the branch locus, and away from its preimage \( F \) restricts to a covering map in the sense of algebraic topology when \( R \) and \( S \) are compact.<sup>[15](https://www.dpmms.cam.ac.uk/~agk22/RS-notes.pdf)</sup> Near any point \( x \in F^{-1}(B) \), the map is expressed in suitable local coordinates by the simple formula \( z \mapsto z^n \), where \( n = v_F(x) \) is the ramification index.<sup>[15](https://www.dpmms.cam.ac.uk/~agk22/RS-notes.pdf)</sup> Every unbranched covering map is a branched cover, by choosing \( B = \varnothing \), and every holomorphic map between Riemann surfaces is a branched cover.<sup>[16](https://www.math.sciences.univ-nantes.fr/~golla/docs/courses/notes_0401.pdf)</sup>

The critical points of such a map form a finite set, since in local holomorphic coordinates a critical point satisfies \( \partial f / \partial z = 0 \).<sup>[2](https://www.math.uci.edu/~jviaclov/courses/218BC_2022.pdf)</sup> The Riemann-Hurwitz formula relates the genera of two surfaces connected by a branched cover and is one of the standard tools of the subject.<sup>[7](https://people.math.osu.edu/cueto.5/teaching/8140/Sp23/index.html)</sup>

## Uniformization

The central classification result is the uniformization theorem: any simply connected Riemann surface is isomorphic to one of three normal forms, the [Riemann sphere](https://www.edgepedia.org/riemann-sphere) \( \widehat{\mathbb{C}} \), the complex plane \( \mathbb{C} \), or the unit disk \( \mathbb{D} \).<sup>[4](https://www.mat.univie.ac.at/~armin/lect/Riemann_surfaces.pdf)</sup> The theorem is a generalization of the [Riemann mapping theorem](https://www.edgepedia.org/riemann-mapping-theorem) in the plane.<sup>[4](https://www.mat.univie.ac.at/~armin/lect/Riemann_surfaces.pdf)</sup> An alternative statement, following uniformization theory, is that the universal covering space of any connected Riemann surface must be one of three possibilities: the unit disk, the complex plane, or the Riemann sphere.<sup>[8](https://www.sciencedirect.com/topics/mathematics/riemann-surface)</sup> The three models are pairwise biholomorphically distinct, so exactly one label applies to each simply connected surface.<sup>[17](https://alphabetamath.cc/complex-analysis/hyperbolic-riemann-surfaces-and-uniformization)</sup> In the Klein-Poincaré form, every Riemann surface \( S \) is conformally equivalent to a quotient \( D/G \), where \( D \) is one of the three canonical domains and \( G \) is a properly discontinuous group of Möbius automorphisms of \( D \).<sup>[18](https://encyclopediaofmath.org/wiki/Uniformization)</sup>

The genus determines the type: a compact Riemann surface of genus \( g = 0 \), \( g = 1 \), or \( g > 1 \) is of elliptic, parabolic, or hyperbolic type respectively.<sup>[14](https://encyclopediaofmath.org/wiki/Riemann_surface)</sup>

## Compact surfaces and algebraic curves

Compact Riemann surfaces occupy a special position because they coincide with algebraic curves. Every compact Riemann surface is isomorphic to a smooth algebraic curve, and conversely a smooth algebraic curve lying in projective space is a compact Riemann surface.<sup>[5](https://math.uchicago.edu/~may/REU2015/REUPapers/Smith.pdf)</sup> The two fields approach the subject from different angles: to an analyst, it is a real two-dimensional surface carrying a [Riemannian metric](https://www.edgepedia.org/riemannian-metric) whose isothermal coordinates induce a holomorphic structure, while an algebraic geometer regards it as a one-dimensional object over the complex numbers.<sup>[8](https://www.sciencedirect.com/topics/mathematics/riemann-surface)</sup> Over the field of complex numbers, every compact Riemann surface corresponds to a smooth projective algebraic curve that is uniquely determined up to isomorphism, although this equivalence does not extend to arbitrary Riemann surfaces.<sup>[19](https://encyclopediaofmath.org/wiki/Genus_of_a_curve)</sup>

On a compact surface the holomorphic functions are scarce: a compact Riemann surface has no holomorphic functions besides the constants.<sup>[6](https://www.mathematik.uni-muenchen.de/~wehler/Lehrveranstaltungen_WS_2019_2020.php)</sup> The working objects are instead meromorphic functions and differential forms, organized through divisors, line bundles, and sheaf cohomology. The Riemann-Roch theorem computes the dimension of spaces of functions and forms with given orders of poles and zeros, and [Serre duality](https://www.edgepedia.org/serre-duality) gives \( \dim_{\mathbb{C}} H^1(X, \Omega) = 1 \) together with the degree formula \( \deg(K) = 2g - 2 \) for any canonical divisor \( K \) on a genus \( g \) compact surface.<sup>[20](https://arxiv.org/abs/1805.06405)</sup><sup> • </sup><sup>[7](https://people.math.osu.edu/cueto.5/teaching/8140/Sp23/index.html)</sup> Further central results include the [Abel map](https://www.edgepedia.org/abel-jacobi-map) and the Jacobian variety, and Abel's theorem.<sup>[20](https://arxiv.org/abs/1805.06405)</sup>

## Moduli spaces and physics

Riemann surfaces of a fixed genus \( g \) with \( n \) marked points vary in families, and the set of isomorphism classes forms the moduli space \( \mathcal{M}_{g,n} \), an object central to mathematics since Riemann's work in the mid-19th century.<sup>[9](https://arxiv.org/abs/2410.13273)</sup> Because surfaces have automorphisms, \( \mathcal{M}_{g,n} \) is not generally a manifold; marking at least three points removes automorphisms in genus 0, and for \( 2g - 2 + n > 0 \) the automorphism groups are finite, making the moduli space an orbifold.<sup>[9](https://arxiv.org/abs/2410.13273)</sup> The Deligne-Mumford compactification, defined by including stable curves, provides a compactification of this space.<sup>[9](https://arxiv.org/abs/2410.13273)</sup>

Physics supplies the most prominent modern applications. As a string travels through spacetime it traces out a Riemann surface, its worldsheet, and the path integrals of string theory are integrals over moduli spaces of Riemann surfaces.<sup>[9](https://arxiv.org/abs/2410.13273)</sup> The moduli space is likewise fundamental to 2D quantum gravity, [topological string theory](https://www.edgepedia.org/topological-string-theory), and matrix models, where Witten's conjecture on intersection theory on moduli space, proved by Kontsevich, connects the physics of matrix integrals with the geometry of \( \mathcal{M}_{g,n} \).<sup>[9](https://arxiv.org/abs/2410.13273)</sup> Riemann surfaces also enter the theory of integrable systems through the algebraic reconstruction method, which finds sections of an eigenvector bundle over a Riemann surface and relates to Baker-Akhiezer and tau functions.<sup>[20](https://arxiv.org/abs/1805.06405)</sup>

In differential geometry, the Weierstrass representation parameterizes minimal surfaces in \( \mathbb{R}^3 \) by holomorphic data on a Riemann surface: a meromorphic function \( g \) and a holomorphic 1-form \( \omega \), with \( \omega \) having a zero of order \( 2m \) exactly where \( g \) has a pole of order \( m \), and every minimal surface in \( \mathbb{R}^3 \) admits such a parameterization.<sup>[21](https://math.uchicago.edu/~dannyc/books/foliations/oupbook.pdf)</sup>

## Computation

Practical work with Riemann surfaces has become a computational field. A compact Riemann surface, a projective algebraic curve, and a Jacobian variety are different views of the same object, and algorithms translate between them: given a Riemann surface, one computes an equation representing it as a plane algebraic curve and a period matrix for it.<sup>[22](https://www.math.fsu.edu/~aluffi/archive/paper35.abs.html)</sup> The period matrix \( \Omega \) describes the analytic Jacobian \( J(C) = \mathbb{C}^g / \Omega \mathbb{Z}^{2g} \) of a curve \( C \) and is approximated numerically by integrating a basis of holomorphic differentials along a homology basis.<sup>[23](https://magma.maths.usyd.edu.au/magma/handbook/text/1497)</sup> Period matrices in turn define the [Riemann theta function](https://www.edgepedia.org/riemann-theta-function), and they connect to number theory through the [Birch and Swinnerton-Dyer conjecture](https://www.edgepedia.org/birch-and-swinnerton-dyer-conjecture), the computation of endomorphism rings of Jacobians, and the Abel-Jacobi map.<sup>[24](https://oops.uni-oldenburg.de/3607/1/neueff18.pdf)</sup>

## References

1. [O. Forster, Riemann Surfaces, course page, LMU München](https://www.math.lmu.de/~forster/lehre/vorlB6s_rie.html)
2. [J. Viaclovsky, Lecture 26, Math 218B, UC Irvine (2022)](https://www.math.uci.edu/~jviaclov/courses/218BC_2022.pdf)
3. [Bernhard Riemann (1826–1866), MacTutor History of Mathematics](https://mathshistory.st-andrews.ac.uk/Biographies/Riemann/)
4. [A. W. Knirsch, Lecture notes on Riemann surfaces, Chapter 27: The Uniformization Theorem, University of Vienna](https://www.mat.univie.ac.at/~armin/lect/Riemann_surfaces.pdf)
5. [N. Smith, Compact Riemann Surfaces: A Threefold Categorical Equivalence, University of Chicago REU (2015)](https://math.uchicago.edu/~may/REU2015/REUPapers/Smith.pdf)
6. [J. Wehler, course page, Mathematisches Institut, LMU München](https://www.mathematik.uni-muenchen.de/~wehler/Lehrveranstaltungen_WS_2019_2020.php)
7. [G. Cueto, Topics in Algebraic Geometry: Riemann Surfaces, Math 8140, Ohio State University, Spring 2023](https://people.math.osu.edu/cueto.5/teaching/8140/Sp23/index.html)
8. [Riemann surface, ScienceDirect Topics overview](https://www.sciencedirect.com/topics/mathematics/riemann-surface)
9. [A. Giacchetto, D. Lewański, Les Houches lecture notes on moduli spaces of Riemann surfaces, arXiv:2410.13273](https://arxiv.org/abs/2410.13273)
10. [Surfaces de Riemann compactes, courbes algébriques complexes et leurs Jacobiennes, arXiv:0903.2156](https://arxiv.org/abs/0903.2156)
11. [B. Riemann, Grundlagen für eine allgemeine Theorie der Functionen einer veränderlichen complexen Grösse (Göttingen, 1851), Trinity College Dublin transcription](https://www.maths.tcd.ie/pub/HistMath/People/Riemann/Grund/)
12. [J. T. Knight, Riemann surfaces of field extensions, Math. Proc. Camb. Phil. Soc. 65 (1969) 635–650](https://www.cambridge.org/core/journals/mathematical-proceedings-of-the-cambridge-philosophical-society/article/riemann-surfaces-of-field-extensions/60DF036951C4FBF4DA0121FFFCBB7AF2)
13. [Riemann's contribution to genus classification, MacTutor, sci_hi](https://mathshistory.st-andrews.ac.uk/SH/riemann_sh.pdf)
14. [Riemann surface, Encyclopedia of Mathematics](https://encyclopediaofmath.org/wiki/Riemann_surface)
15. [A. G. Kovalev, IID Riemann Surfaces notes, Chapter 14: Branched covers, DPMMS, Cambridge](https://www.dpmms.cam.ac.uk/~agk22/RS-notes.pdf)
16. [P. Golla, Chapter 2: Surfaces and the Riemann-Hurwitz theorem, Université de Nantes](https://www.math.sciences.univ-nantes.fr/~golla/docs/courses/notes_0401.pdf)
17. [Hyperbolic Riemann Surfaces and Uniformization, Alphabeta Math](https://alphabetamath.cc/complex-analysis/hyperbolic-riemann-surfaces-and-uniformization)
18. [Uniformization, Encyclopedia of Mathematics](https://encyclopediaofmath.org/wiki/Uniformization)
19. [Genus of a curve, Encyclopedia of Mathematics](https://encyclopediaofmath.org/wiki/Genus_of_a_curve)
20. [B. Eynard, Lectures notes on compact Riemann surfaces, arXiv:1805.06405](https://arxiv.org/abs/1805.06405)
21. [D. Calegari, Foliations and the geometry of 3-manifolds, Chapter 3: Minimal surfaces, University of Chicago](https://math.uchicago.edu/~dannyc/books/foliations/oupbook.pdf)
22. [P. Buser, M. Seppälä, Computing on Riemann Surfaces](https://www.math.fsu.edu/~aluffi/archive/paper35.abs.html)
23. [Magma Handbook, Riemann surfaces, Computational Algebra Group, University of Sydney](https://magma.maths.usyd.edu.au/magma/handbook/text/1497)
24. [C. Neurohr, Efficient integration on Riemann surfaces, University of Oldenburg](https://oops.uni-oldenburg.de/3607/1/neueff18.pdf)

---
*Topic: Encyclopedia › Physical world and mathematics › Mathematics and statistics › Geometry and topology › Complex geometry*

*Initially written Sep 17, 2026 · Reviewed: Sep 30, 2026; Oct 15, 2026 · Edited: Sep 30, 2026; Oct 15, 2026 · Last review: Oct 15, 2026*

*Copyright 2026 EdgeChat AI, a subsidiary of Biostate AI.*

License: Edgepedia Community License 1.0, https://www.edgepedia.org/edgepedia/license