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Riemann sphere

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Model of the extended complex plane plus a point at infinity

The Riemann sphere can be visualized as the complex number plane wrapped around a sphere (by some form ofstereographic projection – details are given below).

Stereographic projection ofcomplex numbers $A {\displaystyle A}$ and $B {\displaystyle B}$ onto the points $\alpha$ and $\beta$ of theRiemann sphere - connect $P(\infty )$ with A or B by a line and determine the intersection with the sphere.

Stereographic projection ofcomplex numbers $A {\displaystyle A}$ and $B {\displaystyle B}$ onto the points $\alpha$ and $\beta$ of theRiemann sphere - connect $P(\infty )$ with A or B by a line and determine the intersection with the sphere.

Inmathematics, theRiemann sphere, named afterBernhard Riemann,^[1] is amodel of theextended complex plane (also called theclosed complex plane): thecomplex plane plus onepoint at infinity. This extended plane represents theextended complex numbers, that is, thecomplex numbers plus a value $\infty$ forinfinity. With the Riemann model, the point $\infty$ is near to very large numbers, just as the point $0 {\displaystyle 0}$ is near to very small numbers.

The extended complex numbers are useful incomplex analysis because they allow fordivision by zero in some circumstances, in a way that makes expressions such as $1/0=\infty$ well-behaved. For example, anyrational function on the complex plane can be extended to aholomorphic function on the Riemann sphere, with thepoles of the rational function mapping to infinity. More generally, anymeromorphic function can be thought of as a holomorphic function whosecodomain is the Riemann sphere.

Ingeometry, the Riemann sphere is the prototypical example of aRiemann surface, and is one of the simplestcomplex manifolds. Inprojective geometry, the sphere is an example of acomplex projective space and can be thought of as the complexprojective line $\mathbf {P} ^{1}(\mathbf {C} )$ , theprojective space of allcomplex lines in $\mathbf {C} ^{2}$ . As with anycompact Riemann surface, the sphere may also be viewed as a projectivealgebraic curve, making it a fundamental example inalgebraic geometry. It also finds utility in other disciplines that depend on analysis and geometry, such as theBloch sphere ofquantum mechanics and in otherbranches of physics.

Extended complex numbers

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The extended complex numbers consist of the complex numbers $\mathbb {C}$ together with $\infty$ . The set of extended complex numbers may be written as $\mathbb {C} \cup \{\infty \}$ , and is often denoted by adding some decoration to the letter $\mathbb {C}$ , such as

{\widehat {\mathbb {C} }},\quad {\overline {\mathbb {C} }},\quad {\text{or}}\quad \mathbb {C} _{\infty }.

The notation $\mathbb {C} ^{*}$ has also seen use, but as this notation is also used for the punctured plane $\mathbb {C} \setminus \{0\}$ , it can lead to ambiguity.^[2]

Geometrically, the set of extended complex numbers is referred to as the Riemann sphere (or extended complex plane).

Arithmetic operations

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Addition of complex numbers may be extended by defining, for $z\in \mathbb {C}$ ,

z+\infty =\infty

andmultiplication may be defined by

z\times \infty =\infty

for all nonzero complex numbers $z {\displaystyle z}$ , with $\infty \times \infty =\infty$ . Note that $\infty +\infty$ , $\infty -\infty$ , and $0\times \infty$ are leftundefined. Unlike the complex numbers, the extended complex numbers do not form afield, since $\infty$ does not have anadditive normultiplicative inverse. Nonetheless, it is customary to definedivision on $\mathbb {C} \cup \{\infty \}$ by

{\frac {z}{0}}=\infty \quad {\text{and}}\quad {\frac {z}{\infty }}=0

for all nonzero complex numbers $z {\displaystyle z}$ with $\infty /0=\infty$ and $0/\infty =0$ . The quotients $0/0$ and $\infty /\infty$ are left undefined.

Rational functions

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Anyrational function $f(z)=g(z)/h(z)$ (in other words, $f(z)$ is the ratio of polynomial functions $g(z)$ and $h(z)$ of $z {\displaystyle z}$ with complex coefficients, such that $g(z)$ and $h(z)$ have no common factor) can be extended to acontinuous function on the Riemann sphere. Specifically, if $z_{0}$ is a complex number such that the denominator $h(z_{0})$ is zero but the numerator $g(z_{0})$ is nonzero, then $f(z_{0})$ can be defined as $\infty$ . Moreover, $f(\infty )$ can be defined as thelimit of $f(z)$ as $z\to \infty$ , which may be finite or infinite.

The set of complex rational functions—whose mathematical symbol is $\mathbb {C} (z)$ —form all possibleholomorphic functions from the Riemann sphere to itself, when it is viewed as aRiemann surface, except for the constant function taking the value $\infty$ everywhere. The functions of $\mathbb {C} (z)$ form an algebraic field, known asthe field of rational functions on the sphere.

For example, given the function

f(z)={\frac {6z^{2}+1}{2z^{2}-50}}

we may define $f(\pm 5)=\infty$ , since the denominator is zero at $\pm 5$ , and $f(\infty )=3$ since $f(z)\to 3$ as $z\to \infty$ . Using these definitions, $f {\displaystyle f}$ becomes a continuous function from the Riemann sphere to itself.

As a complex manifold

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As a one-dimensionalcomplex manifold, the Riemann sphere can be described by twocharts, both with domain equal to the complex number plane $\mathbf {C}$ . Let $\zeta$ be a complex number in one copy of $\mathbf {C}$ , and let $\xi$ be a complex number in another copy of $\mathbf {C}$ . Identify each nonzero complex number $\zeta$ of the first $\mathbf {C}$ with the nonzero complex number $1/\xi$ of the second $\mathbf {C}$ . Then the map

f(z)={\frac {1}{z}}

is called thetransition map between the two copies of $\mathbf {C}$ —the so-called charts—glueing them together. Since the transition maps areholomorphic, they define a complex manifold, called theRiemann sphere. As a complex manifold of 1 complex dimension (i.e. 2 real dimensions), this is also called aRiemann surface.

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 $\zeta$ value and a $\xi$ value, and the two values are related by $\zeta =1/\xi$ . The point where $\xi =0$ should then have $\zeta$ -value " $1/0$ "; in this sense, the origin of the $\xi$ -chart plays the role of $\infty$ in the $\zeta$ -chart. Symmetrically, the origin of the $\zeta$ -chart plays the role of $\infty$ in the $\xi$ -chart.

Topologically, the resulting space is theone-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-definedcomplex structure, so that around every point on the sphere there is a neighborhood that can bebiholomorphically identified with $\mathbf {C}$ .

On the other hand, theuniformization theorem, a central result in the classification of Riemann surfaces, states that everysimply-connected Riemann surface is biholomorphic to the complex plane, thehyperbolic plane, or the Riemann sphere. Of these, the Riemann sphere is the only one that is aclosed surface (acompact surface withoutboundary). Hence the two-dimensional sphere admits a unique complex structure turning it into a one-dimensional complex manifold.

As the complex projective line

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The Riemann sphere can also be defined as thecomplex projective line. The points of the complex projective line can be defined asequivalence classes ofnon-null vectors in the complex vector space $\mathbf {C} ^{2}$ : two non-null vectors $(w,z)$ and $(u,v)$ are equivalent iff $(w,z)=(\lambda u,\lambda v)$ for some non-zero coefficient $\lambda \in \mathbf {C}$ .

In this case, the equivalence class is written $[w,z]$ usingprojective coordinates. Given any point $[w,z]$ in the complex projective line, one of $w {\displaystyle w}$ and $z {\displaystyle z}$ must be non-zero, say $w\neq 0$ . Then by the notion of equivalence, $[w,z]=\left[1,z/w\right]$ , which is in a chart for the Riemann sphere manifold.^[3]

This treatment of the Riemann sphere connects most readily to projective geometry. For example, any line (or smooth conic) in thecomplex projective plane is biholomorphic to the complex projective line. It is also convenient for studying the sphere'sautomorphisms, later in this article.

As a sphere

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

Riemann sphere: almost the whole earth in stereographic azimuthal projection 1:500.000.000 (254 dpi)

The Riemann sphere can be visualized as the unit sphere $x^{2}+y^{2}+z^{2}=1$ in the three-dimensional real space $\mathbf {R} ^{3}$ . To this end, consider thestereographic projection from the unit sphere minus the point $(0,0,1)$ onto the plane $z=0,$ which we identify with the complex plane by $\zeta =x+iy$ . InCartesian coordinates $(x,y,z)$ andspherical coordinates $(\theta ,\varphi )$ on the sphere (with $\theta$ thezenith angle and $\varphi$ theazimuth), the projection is

\zeta ={\frac {x+iy}{1-z}}={\cot }{\bigl (}{\tfrac {1}{2}}\theta {\bigr )}\,e^{i\varphi }.

Similarly, stereographic projection from $(0,0,-1)$ onto the plane $z=0,$ identified with another copy of the complex plane by $\xi =x-iy,$ is written

\xi ={\frac {x-iy}{1+z}}={\tan }{\bigl (}{\tfrac {1}{2}}\theta {\bigr )}\,e^{-i\varphi }.

The inverses of these two stereographic projections are maps from the complex plane to the sphere. The first inverse covers the sphere except the point $(0,0,1)$ , and the second covers the sphere except the point $(0,0,-1)$ . The two complex planes, that are the domains of these maps, are identified differently with the plane $z=0$ , because anorientation-reversal is necessary to maintain consistent orientation on the sphere.

The transition maps between $\zeta$ -coordinates and $\xi$ -coordinates are obtained by composing one projection with the inverse of the other. They turn out to be $\zeta =1/\xi$ and $\xi =1/\zeta$ , as described above. Thus the unit sphere isdiffeomorphic to the Riemann sphere.

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

Metric

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A Riemann surface does not come equipped with any particularRiemannian metric. The Riemann surface's conformal structure does, however, determine a class of metrics: all those whose subordinate conformal structure is the given one. In more detail: The complex structure of the Riemann surface does uniquely determine a metric up toconformal equivalence. (Two metrics are said to be conformally equivalent if they differ by multiplication by a positivesmooth function.) Conversely, any metric on anoriented 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.

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 withconstant curvature in any given conformal class.

In the case of the Riemann sphere, theGauss–Bonnet theorem implies that a constant-curvature metric $\gamma$ must have positivecurvature $K {\displaystyle K}$ . It follows that the metric must beisometric to the sphere of radius $1/{\sqrt {K}}$ in $\mathbf {R} ^{3}$ via stereographic projection. In the $\zeta$ -chart on the Riemann sphere, the metric with $K=1$ is given by

ds^{2}=2\gamma _{\zeta {\overline {\zeta }}}\,d\zeta \,d{\overline {\zeta }}={\frac {4}{\left(1+\zeta {\overline {\zeta }}\right)^{2}}}\,d\zeta \,d{\overline {\zeta }}=\left({\frac {2}{1+|\zeta |^{2}}}\right)^{2}\,|d\zeta |^{2}.

In real coordinates $\zeta =u+iv$ , the formula is

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

Up to a constant factor, this metric agrees with the standardFubini–Study metric on complex projective space (of which the Riemann sphere is an example).The two non-vanishingChristoffel symbols of itsLevi-Civita connection are $\Gamma _{\zeta \zeta }^{\zeta }=-2{\overline {\zeta }}/(1+|\zeta |^{2})$ and its conjugate.This metric is therefore equal to its ownRicci curvature, $\gamma =\mathrm {Ric}$ .

Up to scaling, this is theonly metric on the sphere whose group of orientation-preserving isometries is 3-dimensional (and none is more than 3-dimensional); that group is called ${\mbox{SO}}(3)$ . In this sense, this is by far the most symmetric metric on the sphere. (The group of all isometries, known as ${\mbox{O}}(3)$ , is also 3-dimensional, but unlike ${\mbox{SO}}(3)$ is not a connected space.)

Conversely, let $S {\displaystyle S}$ denote the sphere (as an abstractsmooth ortopological manifold). By the uniformization theorem there exists a unique complex structure on $S {\displaystyle S}$ up to conformal equivalence. It follows that any metric on $S {\displaystyle S}$ is conformally equivalent to theround 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 aconformal manifold, not aRiemannian manifold. However, if one needs to do Riemannian geometry on the Riemann sphere, the round metric is a natural choice (with any fixed radius, though radius $1 {\displaystyle 1}$ is the simplest and most common choice). That is because only a round metric on the Riemann sphere has its isometry group be a 3-dimensional group. (Namely, the group known as ${\mbox{SO}}(3)$ , a continuous ("Lie") group that is topologically the 3-dimensionalprojective space $\mathbf {P} ^{3}$ .)

Automorphisms

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AMöbius transformation acting on the sphere, and on the plane bystereographic projection.

Main article:Möbius transformation

The study of anymathematical object is aided by an understanding of itsgroup 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 invertibleconformal map (i.e. biholomorphic map) from the Riemann sphere to itself. It turns out that the only such maps are theMöbius transformations. These are functions of the form

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

where $a {\displaystyle a}$ , $b {\displaystyle b}$ , $c {\displaystyle c}$ , and $d {\displaystyle d}$ are complex numbers such that $ad-bc\neq 0$ . Examples of Möbius transformations includedilations,rotations,translations, and complex inversion. In fact, any Möbius transformation can be written as a composition of these.

The Möbius transformations arehomographies on the complex projective line. Inprojective coordinates, the transformationf can be written

[\zeta ,\ 1]{\begin{pmatrix}a&c\\b&d\end{pmatrix}}\ =\ [a\zeta +b,\ c\zeta +d]\ =\ {\begin{pmatrix}{\tfrac {a\zeta +b}{c\zeta +d}}&1\end{pmatrix}}\ =\ [f(\zeta ),\ 1].

Thus the Möbius transformations can be described as two-by-two complex matrices with nonzerodeterminant. Since they act on projective coordinates, two matrices yield the same Möbius transformation if and only if they differ by a nonzero factor. Thegroup of Möbius transformations is theprojective linear group ${\mbox{PGL}}(2,\mathbf {C} )$ .

If one endows the Riemann sphere with theFubini–Study metric, then not all Möbius transformations are isometries; for example, the dilations and translations are not. The isometries form a proper subgroup of ${\mbox{PGL}}(2,\mathbf {C} )$ , namely ${\mbox{PSU}}(2)$ . This subgroup is isomorphic to therotation group ${\mbox{SO}}(3)$ , which is the group of symmetries of the unit sphere in $\mathbf {R} ^{3}$ (which, when restricted to the sphere, become the isometries of the sphere).

Applications

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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 $f {\displaystyle f}$ and $g {\displaystyle g}$ . As a map to the complex numbers, it is undefined wherever $g {\displaystyle g}$ 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 is often cited as a construction on which one can easily visualisegeneralised circles,Möbius transformations and conformal maps between connected open subsets of the extended complex plane.

The Riemann sphere has many uses in physics. In quantum mechanics, points on the complex projective line are natural values forphoton polarization states,spin states ofmassive particles of spin $1/2$ , and 2-state particles in general (see alsoQuantum bit andBloch sphere). The Riemann sphere has been suggested as arelativistic model for thecelestial sphere.^[4] Instring theory, theworldsheets of strings are Riemann surfaces, and the Riemann sphere, being the simplest Riemann surface, plays a significant role. It is also important intwistor theory.

Notes

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^Riemann 1857.
^"C^*".Archived from the original on October 8, 2021. RetrievedDecember 12, 2021.
^Goldman 1999, p. 1.
^Penrose 2007, pp. 428–430.

References

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This article includes a list ofgeneral references, butit lacks sufficient correspondinginline citations. Please help toimprove this article byintroducing more precise citations.(August 2010) (Learn how and when to remove this message)

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Brown, James & Churchill, Ruel (1989).Complex Variables and Applications. New York: McGraw-Hill.ISBN 0-07-010905-2.
Goldman, William Mark (1999).Complex Hyperbolic Geometry. Oxford : New York: Oxford University Press.ISBN 0-19-853793-X.
Griffiths, Phillip & Harris, Joseph (1978).Principles of Algebraic Geometry. John Wiley & Sons.ISBN 0-471-32792-1.
Penrose, Roger (2007).The Road to Reality. London: National Geographic Books.ISBN 978-0-679-77631-4.
Riemann, Bernhard (1857)."Theorie der Abel'schen Functionen" [Theory of Abelian functions].Journal für die reine und angewandte Mathematik (in German).54:115–155.
Rudin, Walter (1987).Real and Complex Analysis. New York: McGraw–Hill.ISBN 0-07-100276-6.

External links

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Wikimedia Commons has media related toRiemann sphere.

"Riemann sphere",Encyclopedia of Mathematics,EMS Press, 2001 [1994]
Moebius Transformations Revealed, byDouglas N. Arnold and Jonathan Rogness (a video by two University of Minnesota professors explaining and illustrating Möbius transformations using stereographic projection from a sphere)

Topics inalgebraic curves

Rational curves

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Analytic theory	Elliptic function Elliptic integral Fundamental pair of periods Modular form
Arithmetic theory	Counting points on elliptic curves Division polynomials Hasse's theorem on elliptic curves Mazur's torsion theorem Modular elliptic curve Modularity theorem Mordell–Weil theorem Nagell–Lutz theorem Supersingular elliptic curve Schoof's algorithm Schoof–Elkies–Atkin algorithm
Applications	Elliptic curve cryptography Elliptic curve primality

Higher genus

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Divisors on curves	Abel–Jacobi map Brill–Noether theory Clifford's theorem on special divisors Gonality of an algebraic curve Jacobian variety Riemann–Roch theorem Weierstrass point Weil reciprocity law
Moduli	ELSV formula Gromov–Witten invariant Hodge bundle Moduli of algebraic curves Stable curve
Morphisms	Hasse–Witt matrix Riemann–Hurwitz formula Prym variety Weber's theorem (Algebraic curves)
Singularities	A_k singularity Acnode Crunode Cusp Delta invariant Tacnode
Vector bundles	Birkhoff–Grothendieck theorem Stable vector bundle Vector bundles on algebraic curves

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