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Encyclopedia > Riemann sphere
The Riemann sphere can be visualized as the complex number plane wrapped around a sphere (by some form of stereographic projection — details are given below).

In mathematics, the Riemann sphere is a way of extending the plane of complex numbers with one additional point at infinity, in a way that makes expressions such as Image File history File links No higher resolution available. ... Image File history File links No higher resolution available. ... For other meanings of mathematics or uses of math and maths, see Mathematics (disambiguation) and Math (disambiguation). ... This article is about the mathematical construct. ... In mathematics, a complex number is a number which is often formally defined to consist of an ordered pair of real numbers , often written: In mathematics, the adjective complex means that the underlying number field is complex numbers, for example complex analysis, complex matrix, complex polynomial and complex Lie algebra. ... The point at infinity, also called ideal point, is a point which when added to the real number line yields a closed curve called the real projective line, . Nota Bene: The real projective line is not equivalent to the extended real number line. ...

1 / 0 = infty

well-behaved and useful, at least in certain contexts. It is named after 19th century mathematician Bernhard Riemann. It is also called the complex projective line, denoted mathbb{CP}^1. Alternative meaning: Nineteenth Century (periodical) (18th century — 19th century — 20th century — more centuries) As a means of recording the passage of time, the 19th century was that century which lasted from 1801-1900 in the sense of the Gregorian calendar. ... Bernhard Riemann. ... In mathematics, a projective line is a one-dimensional projective space. ...


On a purely algebraic level, the complex numbers with an extra infinity element constitute a number system known as the extended complex numbers. Arithmetic with infinity does not obey all of the usual rules of algebra, and so the extended complex numbers do not form a field. However, the Riemann sphere is geometrically and analytically well-behaved, even near infinity; it is a one-dimensional complex manifold, also called a Riemann surface. This article is about the branch of mathematics. ... In abstract algebra, a field is an algebraic structure in which the operations of addition, subtraction, multiplication, and division (except division by zero) may be performed and the associative, commutative, and distributive rules hold, which are familiar from the arithmetic of ordinary numbers. ... 2-dimensional renderings (ie. ... In differential geometry, a complex manifold is a manifold such that every neighborhood looks like the complex n-space in a coherent way. ... 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. ...


In complex analysis, the Riemann sphere facilitates an elegant theory of meromorphic functions. The Riemann sphere is ubiquitous in projective geometry and algebraic geometry as a fundamental example of a complex manifold, projective space, and algebraic variety. It also finds utility in other disciplines that depend on analysis and geometry, such as quantum mechanics and other branches of physics. Plot of the function f(x)=(x2-1)(x-2-i)2/(x2+2+2i). ... 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. ... Projective geometry is a non-metrical form of geometry. ... Algebraic geometry is a branch of mathematics which, as the name suggests, combines techniques of abstract algebra, especially commutative algebra, with the language and the problematics of geometry. ... This article does not cite its references or sources. ... In mathematics, an algebraic variety is essentially a set of common zeroes of a set of polynomials. ... For a less technical and generally accessible introduction to the topic, see Introduction to quantum mechanics. ... A magnet levitating above a high-temperature superconductor demonstrates the Meissner effect. ...

Contents

As a complex manifold

As a one-dimensional complex manifold, the Riemann sphere can be described by two charts, both with domain equal to the complex number plane mathbb{C}. Let ζ and ξ be complex coordinates on mathbb{C}. Identify the nonzero complex numbers ζ with the nonzero complex numbers ξ using the transition maps In topology, an atlas describes how a complicated space is glued together from simpler pieces. ...

ζ = 1 / ξ,
ξ = 1 / ζ.

Since the transition maps are holomorphic, they define a complex manifold, called the Riemann sphere. 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. ...


Intuitively, the transition maps indicate how to glue two planes together to form the Riemann sphere. The planes are glued in an "inside-out" manner, so that they overlap almost everywhere, with each plane contributing just one point (its origin) missing from the other plane. In other words, (almost) every point in the Riemann sphere has both a ζ value and a ξ value, and the two values are related by ζ = 1 / ξ. The point where ξ = 0 should then have ζ-value "1 / 0"; in this sense, the origin of the ξ-chart plays the role of "infty" in the ζ-chart. Symmetrically, the origin of the ζ-chart plays the role of infty with respect to the ξ-chart.


Topologically, the resulting space is the one-point compactification of a plane into the sphere. However, the Riemann sphere is not merely a topological sphere. It is a sphere with a well-defined complex structure, so that around every point on the sphere there is a neighborhood that can be biholomorphically identified with mathbb{C}. On the other hand, the two-dimensional sphere admits a unique complex structure turning it into a one-dimensional complex manifold. A Möbius strip, an object with only one surface and one edge; such shapes are an object of study in topology. ... In mathematics, compactification is the process or result of enlarging a topological space to make it compact. ... In differential geometry, a complex manifold is a manifold such that every neighborhood looks like the complex n-space. ...


The uniformization theorem, a central result in the classification of Riemann surfaces, states that the only simply-connected one-dimensional complex manifolds are the complex plane, the hyperbolic plane, and the Riemann sphere. Of these, the Riemann sphere is the only one that is closed (compact and without boundary). In mathematics, the uniformization theorem for surfaces says that any surface admits a Riemannian metric of constant Gauss curvature. ... 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. ... A triangle immersed in a saddle-shape plane, as well as two diverging parallel lines. ... In mathematics, a closed manifold, or compact manifold, is a manifold that is compact as a topological space. ... In mathematics, a subset of Euclidean space Rn is called compact if it is closed and bounded. ... In topology, the boundary of a subset S of a topological space X is the set of points which can be approached both from S and from the outside of S. More formally, it is the set of points in the closure of S, not belonging to the interior of...


As the complex projective line

The Riemann sphere can also be defined as the complex projective line. This is the subset of mathbb{C}^2 consisting of all pairs (α,β) of complex numbers, not both zero, modulo the equivalence relation For quotient spaces in linear algebra, see quotient space (linear algebra). ... In mathematics, an equivalence relation is a binary relation between two elements of a set which groups them together as being equivalent in some way. ...

(α,β) = (λα,λβ)

for all nonzero complex numbers λ. The complex plane mathbb{C}, with coordinate ζ, can be mapped into the complex projective line by

(α,β) = (ζ,1).

Another copy of mathbb{C} with coordinate ξ can be mapped in by

(α,β) = (1,ξ).

These two complex charts cover the projective line. For nonzero ξ the identifications

(1,ξ) = (1 / ξ,1) = (ζ,1)

demonstrate that the transition maps are ζ = 1 / ξ and ξ = 1 / ζ, as above.


This treatment of the Riemann sphere connects most readily to projective geometry. For example, any line (or smooth conic) in the complex projective plane is biholomorphic to the complex projective line. It is also convenient for studying the sphere's automorphisms, later in this article. In mathematics, the complex projective plane, usually denoted CP2, is the two-dimensional complex projective space. ... In mathematics, an automorphism is an isomorphism from a mathematical object to itself. ...


As a sphere

Stereographic projection of a complex number A onto a point α of the Riemann sphere.
Stereographic projection of a complex number A onto a point α of the Riemann sphere.

The Riemann sphere can be visualized as the unit sphere x2 + y2 + z2 = 1 in the three-dimensional real space mathbb{R}^3. To this end, consider the stereographic projection from the unit sphere minus the point (0,0,1) onto the plane z = 0, which we identify with the complex plane by ζ = x + iy. In Cartesian coordinates (x,y,z) and spherical coordinates (φ,θ) on the sphere (with φ the zenith and θ the azimuth), the projection is Image File history File links Metadata Size of this preview: 750 × 600 pixelsFull resolution‎ (1,280 × 1,024 pixels, file size: 215 KB, MIME type: image/jpeg) File historyClick on a date/time to view the file as it appeared at that time. ... Image File history File links Metadata Size of this preview: 750 × 600 pixelsFull resolution‎ (1,280 × 1,024 pixels, file size: 215 KB, MIME type: image/jpeg) File historyClick on a date/time to view the file as it appeared at that time. ... Stereographic projection of a circle of radius R onto the x axis. ... Cartesian means relating to the French mathematician and philosopher Descartes, who, among other things, worked to merge algebra and Euclidean geometry. ... This article describes some of the common coordinate systems that appear in elementary mathematics. ...

zeta = frac{x + i y}{1 - z} = cot(phi / 2) ; e^{i theta}.

Similarly, stereographic projection from (0,0, − 1) onto the z = 0 plane, identified with another copy of the complex plane by ξ = xiy, is written

&# 0;= frac{x - i y}{1 + z} = tan(phi / 2) ; e^{-i theta}.

(The two complex planes are identified differently with the plane z = 0. An orientation-reversal is necessary to maintain consistent orientation on the sphere, and in particular complex conjugation causes the transition maps to be holomorphic.) The transition maps between ζ-coordinates and ξ-coordinates are obtained by composing one projection with the inverse of the other. They turn out to be ζ = 1 / ξ and ξ = 1 / ζ, as described above. Thus the unit sphere is diffeomorphic to the Riemann sphere. In mathematics, an orientation on a real vector space is a choice of which ordered bases are positively oriented (or right-handed) and which are negatively oriented (or left-handed). ... In mathematics, a diffeomorphism is a kind of isomorphism of smooth manifolds. ...


Under this diffeomorphism, the unit circle in the ζ-chart, the unit circle in the ξ-chart, and the equator of the unit sphere are all identified. The unit disk | ζ | < 1 is identified with the southern hemisphere z < 0, while the unit disk | ξ | < 1 is identified with the northern hemisphere z > 0.


Metric

A Riemann surface does not come equipped with any particular Riemannian metric. However, the complex structure of the Riemann surface does uniquely determine a metric up to conformal equivalence. (Two metrics are said to be conformally equivalent if they differ by multiplication by a positive smooth function.) Conversely, any metric on an oriented surface uniquely determines a complex structure, which depends on the metric only up to conformal equivalence. Complex structures on an oriented surface are therefore in one-to-one correspondence with conformal classes of metrics on that surface. In mathematics, in Riemannian geometry, the metric tensor is a tensor of rank 2 that is used to measure distance and angle in a space. ... In mathematics and theoretical physics, two geometries are conformally equivalent if there exists a conformal transformation (an angle-preserving transformation) that maps one geometry to the other one. ... In mathematical analysis, a differentiability class is a classification of functions according to the properties of their derivatives. ... In mathematics, an orientation on a real vector space is a choice of which ordered bases are positively oriented and which are negatively oriented. ...


Within a given conformal class, one can use conformal symmetry to find a representative metric with convenient properties. In particular, there is always a complete metric with constant curvature in any given conformal class. In mathematics, constant curvature in differential geometry is a concept most commonly applied to surfaces. ...


In the case of the Riemann sphere, the Gauss-Bonnet theorem implies that a constant-curvature metric must have positive curvature K. It follows that the metric must be isometric to the sphere of radius 1 / sqrt K in mathbb{R}^3 via stereographic projection. In the ζ-chart on the Riemann sphere, the metric with K = 1 is given by 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). ... Curvature is the amount by which a geometric object deviates from being flat. ... In the study of Riemannian geometry in mathematics, a local isometry from one (pseudo-)Riemannian manifold to another is a map which pulls back the metric tensor on the second manifold to the metric tensor on the first. ...

ds^2 = left(frac{2}{1+|zeta|^2}right)^2,|dzeta|^2 = frac{4}{left(1 + zeta bar zetaright)^2},dzeta dbar zeta.

In real coordinates ζ = u + iv, the formula is

ds^2 = frac{4}{left(1 + u^2 + v^2right)^2} left(du^2 + dv^2right).

Up to a constant factor, this metric agrees with the standard Fubini-Study metric on complex projective space (of which the Riemann sphere is an example). In mathematics, a Kähler 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. ...


Conversely, let S denote the sphere (as an abstract smooth or topological manifold). By the uniformization theorem there exists a unique complex structure on S. It follows that any metric on S is conformally equivalent to the round metric. All such metrics determine the same conformal geometry. The round metric is therefore not intrinsic to the Riemann sphere, since "roundness" is not an invariant of conformal geometry. The Riemann sphere is only a conformal manifold not a Riemannian manifold. However, if one needs to do Riemannian geometry on the Riemann sphere, the round metric is a natural choice. Informally, a differentiable manifold is a type of manifold (which is in turn a kind of topological space) that is locally similar enough to Euclidean space to allow one to do calculus. ... In mathematics, a manifold M is a type of space, characterized in one of two equivalent ways: near every point of the space, we have a coordinate system; or near every point, the environment is like that in Euclidean space of a given dimension. ...


Automorphisms

A Möbius transformation acting on the sphere, and on the plane by stereographic projection.
A Möbius transformation acting on the sphere, and on the plane by stereographic projection.

The study of any mathematical object is aided by an understanding of its group of automorphisms, meaning the maps from the object to itself that preserve the essential structure of the object. In the case of the Riemann sphere, an automorphism is an invertible biholomorphic map from the Riemann sphere to itself. It turns out that the only such maps are the Möbius transformations. These are functions of the form Image File history File links Mob3d-elip-opp-200. ... Image File history File links Mob3d-elip-opp-200. ... In mathematics, a Möbius transformation is a bijective conformal mapping of the extended complex plane (i. ... This picture illustrates how the hours on a clock form a group under modular addition. ... In mathematics, a Möbius transformation is a bijective conformal mapping of the extended complex plane (i. ...

f(zeta) = frac{a zeta + b}{c zeta + d},

where a, b, c, and d are complex numbers such that a d - b c neq 0. Examples of Möbius transformations include dilations, rotations, translations, and complex inversion. In fact, any Möbius transformation can be written as a composition of these. Dilation in physiological context may mean: pupil dilation (mydriasis) dilation of blood vessels (vasodilation) cervical dilation (or dilation of the cervix) in childbirth Dilation and curettage (surgical dilation) In mathematics: Dilation This is a disambiguation page &#8212; a navigational aid which lists other pages that might otherwise share the same... This article is about rotation as a movement of a physical body. ... In Euclidean geometry, translation is a transformation of Euclidean space which moves every point by a fixed distance in the same direction. ...


The Möbius transformations are profitably viewed as transformations on the complex projective line. In projective coordinates, the transformation f can be written

f(alpha, beta) = (a alpha + b beta, c alpha + d beta) = begin{pmatrix} alpha & beta end{pmatrix} begin{pmatrix} a & c  b & d end{pmatrix}.

Thus the Möbius transformations can be described as 2 times 2 complex matrices with nonzero determinant; two matrices yield the same Möbius transformation if and only if they differ by a nonzero constant. Thus the Möbius transformations exactly correspond to the projective linear transformations mathrm{PGL}_2(mathbb{C}). In algebra, a determinant is a function depending on n that associates a scalar, det(A), to every n×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. ... In mathematics, a linear transformation (also called linear map or linear operator) is a function between two vector spaces that preserves the operations of vector addition and scalar multiplication. ...


If one endows the Riemann sphere with the Fubini-Study metric, then not all Möbius transformations are isometries; for example, the dilations and translations are not. The isometries form a proper subgroup of mathrm{PGL}_2(mathbb{C}), namely PSU2. This subgroup is isomorphic to the rotation group SO(3), which is the isometry group of the unit sphere in mathbb{R}^3. In mechanics and geometry, the rotation group is the set of all rotations about the origin of 3-dimensional Euclidean space, R3. ...


Applications

In complex analysis, a meromorphic function on the complex plane (or on any Riemann surface, for that matter) is a ratio f / g of two holomorphic functions f and g. As a map to the complex numbers, it is undefined wherever g is zero. However, it induces a holomorphic map (f,g) to the complex projective line that is well-defined even where g = 0. This construction is helpful in the study of holomorphic and meromorphic functions. For example, on a compact Riemann surface there are no non-constant holomorphic maps to the complex numbers, but holomorphic maps to the complex projective line are abundant.


The Riemann sphere has many uses in physics. In quantum mechanics, points on the complex projective line are natural values for photon polarization states, spin states of massive particles of spin 1/2, and 2-state particles in general. The Riemann sphere has been suggested as a relativistic model for the celestial sphere. In string theory, the worldsheets of strings are Riemann surfaces, and the Riemann sphere, being the simplest Riemann surface, plays a significant role. It is also important in twistor theory. In modern physics the photon is the elementary particle responsible for electromagnetic phenomena. ... In electrodynamics, polarization (also spelled polarisation) is the property of electromagnetic waves, such as light, that describes the direction of their transverse electric field. ... In physics, spin refers to the angular momentum intrinsic to a body, as opposed to orbital angular momentum, which is the motion of its center of mass about an external point. ... For other uses, see Mass (disambiguation). ... Helium atom (schematic) Showing two protons (red), two neutrons (green) and two electrons (yellow). ... For a less technical and generally accessible introduction to the topic, see Introduction to general relativity. ... The celestial sphere is divided by the celestial equator. ... Interaction in the subatomic world: world lines of pointlike particles in the Standard Model or a world sheet swept up by closed strings in string theory String theory is a model of fundamental physics, whose building blocks are one-dimensional extended objects called strings, rather than the zero-dimensional point... A world line of an object or person is the sequence of events labeled with time and place, that marks the history of the object or person. ... The twistor theory, originally developed by Roger Penrose in 1967, is the mathematical theory which maps the geometric objects of the four dimensional space-time (Minkowski space) into the geometric objects in the 4-dimensional complex space with the metric signature (2,2). ...


References

  • Brown, James and Churchill, Ruel (1989). Complex Variables and Applications. New York: McGraw-Hill. ISBN 0070109052. 
  • Griffiths, Phillip and Harris, Joseph (1978). Principles of Algebraic Geometry. John Wiley & Sons. ISBN 0-471-32792-1. 
  • Penrose, Roger (2005). The Road to Reality. New York: Knopf. ISBN 0-679-45443-8. 
  • Rudin, Walter (1987). Real and Complex Analysis. New York: McGraw-Hill. ISBN 0071002766. 

See also

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, the cross-ratio cr( w, x, y, z ) of an ordered quadruple of complex numbers (which may be real numbers) is Cross-ratios are preserved by linear fractional transformations, i. ... In mathematics, the Hopf bundle (or Hopf fibration), named after Heinz Hopf, is an important example of a fiber bundle. ... In mathematics, a dessin denfant (French for a childs drawing) is a connected graph with a cyclic ordering of edges at each vertex, and each vertex being colored black or white and with no edge having endpoints of the same color. ...

External links

  • Twistor Theory, by R. Penrose and F. Hadrovich
  • Moebius Transformations Revealed, by Douglas N. Arnold and Jonathan Rogness (a video by two University of Minnesota professors explaining and illustrating Möbius transformations using stereographic projection from a sphere)

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