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Encyclopedia > Riemann surface
Riemann surface for the function f(z) = sqrt(z)

In mathematics, particularly in complex analysis, a Riemann surface, 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. Image File history File links Riemann_sqrt. ... Image File history File links Riemann_sqrt. ... For other meanings of mathematics or math, see mathematics (disambiguation). ... Complex analysis is the branch of mathematics investigating functions of complex numbers, and is of enormous practical use in many branches of mathematics, including applied mathematics. ... Bernhard Riemann. ... In differential geometry, a complex manifold is a manifold such that every neighborhood looks like the complex n-space. ... In mathematics, the complex plane is a way of visualising the space of the complex numbers. ... A MÃ¶bius strip, a surface with only one side and one edge; such shapes are an object of study in topology. ... A sphere (< Greek ÏƒÏ†Î±Î¯ÏÎ±) is a perfectly symmetrical geometrical object. ... A torus. ...

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 or the logarithm. Holomorphic functions are the central object of study of complex analysis; they are functions defined on an open subset of the complex number plane C with values in C that are complex-differentiable at every point. ... This diagram does not represent a true function; because the element 3, in X, is associated with two elements b and c, in Y. In mathematics, a multivalued function is a total relation; i. ... In mathematics, a square root of a number x is a number whose square (the result of multiplying the number by itself) is x. ... The natural logarithm, formerly known as the hyperbolic logarithm, is the logarithm to the base e, where e is equal to 2. ...

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. On a sphere, the sum of the angles of a triangle is not equal to 180Â°. A sphere is not a Euclidean space, but locally the laws of the Euclidean geometry are good approximations. ... An open surface with X-, Y-, and Z-contours shown. ... In differential geometry, a complex manifold is a manifold such that every neighborhood looks like the complex n-space. ... This article or section should be merged with Orientable manifold. ... A MÃ¶bius strip made with a piece of paper and tape. ... The Klein bottle immersed in three-dimensional space. ... Projective plane - Wikipedia, the free encyclopedia /**/ @import /skins-1. ...

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. In mathematics, specifically in complex analysis and algebraic geometry, the Riemann-Roch theorem is an important tool in the computation of the dimension of the space of meromorphic functions with prescribed zeroes and allowed poles. ...

## Contents

Let X be a Hausdorff space. A homeomorphism from an open subset UX to a subset of C is called a chart. Two charts f and g whose domains intersect are said to be compatible if the maps f o g−1 and g o f −1 are holomorphic over their domains. If A is a collection of compatible charts and if any x in X is in the domain of some f in A, then we say that A is an atlas. When we endow X with an atlas A, we say that (X, A) is a Riemann surface. If the atlas is understood, we simply say that X is a Riemann surface. In topology and related branches of mathematics, a Hausdorff space is a topological space in which points can be separated by neighbourhoods. ... In the mathematical field of topology a homeomorphism or topological isomorphism (from the Greek words homeos = identical and morphe = shape) is a special isomorphism between topological spaces which respects topological properties. ... In topology, an atlas describes how a complicated space is glued together from simpler pieces. ... Holomorphic functions are the central object of study of complex analysis; they are functions defined on an open subset of the complex number plane C with values in C that are complex-differentiable at every point. ... In topology, an atlas describes how a complicated space is glued together from simpler pieces. ...

Different atlases can give rise to essentially the same Riemann surface structure on X; to avoid this ambiguity, one sometimes demands that the given atlas on X be maximal, in the sense that it is not contained in any other atlas. Every atlas A is contained in a unique maximal one by Zorn's lemma. Zorns lemma, also known as the Kuratowski-Zorn lemma, is a proposition of set theory that states: Every non-empty partially ordered set in which every chain (i. ...

## Examples

• The complex plane C is perhaps the most trivial 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. Important examples of non-compact Riemann surfaces are provided by analytic continuation (see below.)

In mathematics, the complex plane is a way of visualising the space of the complex numbers. ... In mathematics, the complex conjugate of a complex number is given by changing the sign of the imaginary part. ... In topology and related fields of mathematics, a set U is called open if, intuitively speaking, you can wiggle or change any point x in U by a small amount in any direction and still be inside U. In other words, if x is surrounded only by elements of U... A rendering of the Riemann Sphere. ... In mathematics, a subset of Euclidean space Rn is called compact if it is closed and bounded. ... In algebraic geometry, an algebraic curve is an algebraic variety of dimension equal to 1. ...

## Properties and further definitions

A function f : MN 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 conformally equivalent 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. Partial plot of a function f. ... In topology, an atlas describes how a complicated space is glued together from simpler pieces. ... In mathematics, a bijection, bijective function, or one-to-one correspondence is a function that is both injective (one-to-one) and surjective (onto), and therefore bijections are also called one_to_one and onto. ...

Every simply connected Riemann surface is conformally equivalent to C or to the Riemann sphere C ∪ {∞} or to the open disk {zC : |z| < 1}. This is the main step in the uniformization theorem. A geometrical object is called simply connected if it consists of one piece and doesnt have any circle-shaped holes or handles. Higher-dimensional holes are allowed. ... In mathematics, the uniformization theorem for surfaces says that any surface admits a Riemannian metric of constant Gauss curvature. ...

The uniformization theorem states that every connected Riemann surface admits a unique complete 2-dimensional real Riemann metric with constant curvature -1, 0 or 1. The Riemann surfaces with curvature -1 are called hyperbolic; the open disk with the Poincar�-metric of constant curvature -1 is the local model. Examples include all surfaces with genus g>1. The Riemann surfaces with curvature 0 are called parabolic; C and the 2-torus are typical parabolic Riemann surfaces. Finally, the surfaces with curvature +1 are known as elliptic; the Riemann sphere C ∪ {∞} is the only example. In mathematical analysis, a metric space M is said to be complete (or Cauchy) if every Cauchy sequence of points in M has a limit that is also in M. Intuitively, a space is complete if it doesnt have any holes, if there arent any points missing. For... In Riemannian geometry, a Riemannian manifold is a real differentiable manifold in which each tangent space is equipped with an inner product in a manner which varies smoothly from point to point. ... Curvature refers to a number of loosely related concepts in different areas of geometry. ... A rendering of the Riemann Sphere. ...

For every closed parabolic Riemann surface, the fundamental group is isomorphic to a rank 2 lattice, and thus the surface can be constructed as C/Γ, where C is the complex plane and Γ is the lattice. The set of representatives of the cosets are called fundamental domains. In mathematics, the fundamental group is one of the basic concepts of algebraic topology. ... In mathematics, especially in geometry and group theory, a lattice in Rn is a discrete subgroup of Rn which spans the real vector space Rn. ... In mathematics, given a lattice &#915; in a Lie group G, a fundamental domain is a set D of representatives for the cosets G/&#915;, that is also a well-behaved set topologically, in a sense that can be made precise in one of several ways. ...

Similarly, 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. In mathematics, a Fuchsian group is a particular type of group of isometries of the hyperbolic plane. ... In mathematics, a Fuchsian model is a representation of a hyperbolic Riemann surface R. By the uniformization theorem, every Riemann surface is either elliptic, parabolic or hyperbolic. ... In mathematics, the upper half plane H is the set of complex numbers x + iy such that y > 0. ... In mathematics, a free regular set is a subset of a topological space that is acted upon disjointly under a given group action. ... In mathematics, each closed surface in the sense of geometric topology can be constructed from an even-sided oriented polygon, called a fundamental polygon, by pairwise identification of its edges. ...

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. In mathematics, the genus has few different meanings Topology The genus of a connected, oriented surface is an integer representing the maximum number of cuttings along closed simple curves without rendering the resultant manifold disconnected. ... The Gauss-Bonnet theorem in differential geometry is an important statement about surfaces which connects their geometry (in the sense of curvature) to their topology (in the sense of the Euler characteristic). ...

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 R2 to R2 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. This article or section should be merged with Orientable manifold. ... In vector calculus, the Jacobian is shorthand for either the Jacobian matrix or its determinant, the Jacobian determinant. ... In algebra, a determinant is a function depending on n that associates a scalar det(A) to every nÃ—n square matrix A. The fundamental geometric meaning of a determinant is as the scale factor for volume when A is regarded as a linear transformation. ...

## Other Examples

• As noted above, the Riemann sphere is the only elliptic Riemann surface.
• The only parabolic, simply connected Riemann surface is the complex plane. All parabolic surfaces can be obtained as a quotient of the plane. All parabolic surfaces are homeomorphic to either the plane, the annulus, or the torus. However it does not follow that all tori are biholomorphic to each other. This is the first appearance of the problem of moduli. The modulus of a toris can be captured by a single complex number with positive imaginary part. In fact, the marked moduli space (Teichmuller space) of the torus is biholomorphic to the open unit disk.
• The only hyperbolic, simply connected Riemann surface is the open unit disk. The celebrated Riemann Mapping Theorem states that any simply connected strict subset of the complex plane is biholomorphic to the unit disk. All hyperbolic surfaces are quotients of the unit disk. 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 Teichmuller space. The problem of moduli (solved by Ahlfors and extended by Bers) was to justify Riemann's claim that for a closed surface of genus g, 3g - 3 complex parameters suffice.

In mathematics, given a Riemann surface X, the TeichmÃ¼ller space of X, notated TX or Teich(X), is a complex manifold whose points represent all complex structures of Riemann surfaces whose underlying topological structure is the same as that of X. It is named after the German mathematician Oswald... The Cantor set, introduced by German mathematician Georg Cantor, is a remarkable construction involving only the real numbers between zero and one. ...

## Functions

Every non-compact Riemann surface admits non-constant holomorphic functions (with values in $mathbb{C}$). In fact, every non-compact Riemann surface is a Stein manifold. In mathematics, a Stein manifold in the theory of several complex variables and complex manifolds is a closed, complex submanifold of the vector space of n complex dimensions. ...

In contrast, on a compact Riemann surface every holomorphic function with value in $mathbb{C}$ is constant due to the maximum principle. However, there always exists non-constant meromorphic functions (=holomorphic functions with values in the Riemann sphere C ∪ {∞}). A meromorphic function is a function that is holomorphic on an open subset of the complex number plane C (or on some other connected Riemann surface) except at points in a set of isolated poles, which are certain well-behaved singularities. ... A rendering of the Riemann Sphere. ...

## History

Riemann surfaces were first studied by Bernhard Riemann and were named after him. Bernhard Riemann. ...

## In art and literature

• 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.

Hand with Reflecting Sphere (Self-Portrait in Spherical Mirror), 1935. ... This article or section may contain original research or unverified claims. ... Brave New World is a dystopian novel by Aldous Huxley, first published in 1932. ...

Algebraic geometry is a branch of mathematics which, as the name suggests, combines abstract algebra, especially commutative algebra, with geometry. ... In mathematics, conformal geometry is the study of the set of angle-preserving (conformal) transformations on a Euclidean-like space with a point added at infinity, or a Minkowski-like space with a couple of points added at infinity. That is, the setting is a compactification of a familiar space... In mathematics, a KÃ¤hler manifold is a complex manifold which also carries a Riemannian metric and a symplectic form on the underlying real manifold in such a way that the three structures (complex, Riemannian, and symplectic) are all mutually compatible. ... In mathematics, given a Riemann surface X, the TeichmÃ¼ller space of X, notated TX or Teich(X), is a complex manifold whose points represent all complex structures of Riemann surfaces whose underlying topological structure is the same as that of X. // Relation to moduli space The TeichmÃ¼ller space... A rendering of the Riemann Sphere. ... In mathematics, the branching theorem is a theorem about Riemann surfaces. ... In mathematics, Hurwitzs automorphisms theorem bounds the group of automorphisms, via conformal mappings, of a compact Riemann surface of genus g > 1, telling us that the order of the group of such automorphisms is bounded by 84(g &#8722; 1). ... In mathematics, the identity theorem for Riemann surfaces is a theorem that states that a holomorphic function is completely determined by its values on any subset of its domain that has a limit point. ... In mathematics, specifically in complex analysis and algebraic geometry, the Riemann-Roch theorem is an important tool in the computation of the dimension of the space of meromorphic functions with prescribed zeroes and allowed poles. ... In mathematics, the Riemann-Hurwitz formula describes the relationship of the Euler characteristics of two surfaces when one is a ramified covering of the other. ... The Riemann mapping theorem in complex analysis states the following: if U is a simply connected open subset of the complex number plane C which is not all of C, then there exists a bijective holomorphic conformal map f : U -> D, where D = { z in C : |z| < 1 } denotes the... In mathematics, the uniformization theorem for surfaces says that any surface admits a Riemannian metric of constant Gauss curvature. ...

## References

• Hershel M. Farkas and Irwin Kra, Riemann Surfaces (1980), Springer-Verlag, New York. ISBN 0-387-90465-4
• Jürgen Jost, Compact Riemann Surfaces (2002), Springer-Verlag, New York. ISBN 3-540-43299-X
• Riemann Surface on PlanetMath

Results from FactBites:

 PlanetMath: Riemann surface (1824 words) The simplest example of a Riemann surface which is not a subset of the complex plane is the Riemann sphere. A Riemann surface in the narrower sense is a branched covering of the complex plane. When one has constructed this surface and convinced oneself that it has the topology of a torus, one is well on one's way to developing an intuitive understanding of Riemann surfaces.
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