Riemann sphere

Last updated
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). RiemannKugel.svg
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, named after Bernhard Riemann, [1] 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.


The extended complex numbers are useful in complex analysis because they allow for division by zero in some circumstances, in a way that makes expressions such as well-behaved. For example, any rational function on the complex plane can be extended to a holomorphic function on the Riemann sphere, with the poles of the rational function mapping to infinity. More generally, any meromorphic function can be thought of as a holomorphic function whose codomain is the Riemann sphere.

In geometry, the Riemann sphere is the prototypical example of a Riemann surface, and is one of the simplest complex manifolds. In projective geometry, the sphere can be thought of as the complex projective line P1(C), the projective space of all complex lines in C2. As with any compact Riemann surface, the sphere may also be viewed as a projective algebraic curve, making it a fundamental example in algebraic geometry. It also finds utility in other disciplines that depend on analysis and geometry, such as the Bloch sphere of quantum mechanics and in other branches of physics.

The extended complex plane is also called the closed complex plane.

Extended complex numbers

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

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

Arithmetic operations

Addition of complex numbers may be extended by defining, for zC,

for any complex number z, and multiplication may be defined by

for all nonzero complex numbers z, with ∞ × ∞ = ∞. Note that and 0 × ∞ are left undefined. Unlike the complex numbers, the extended complex numbers do not form a field, since does not have a multiplicative inverse. Nonetheless, it is customary to define division on C ∪ {∞} by

for all nonzero complex numbers z with /0 = ∞ and 0/ = 0. The quotients 0/0 and / are left undefined.

Rational functions

Any rational 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 with complex coefficients, such that g(z) and h(z) have no common factor) can be extended to a continuous function on the Riemann sphere. Specifically, if z0 is a complex number such that the denominator h(z0) is zero but the numerator g(z0) is nonzero, then f(z0) can be defined as ∞. Moreover, f(∞) can be defined as the limit of f(z) as z → ∞, which may be finite or infinite.

The set of complex rational functions — whose mathematical symbol is C(z) — form all possible holomorphic functions from the Riemann sphere to itself, when it is viewed as a Riemann surface, except for the constant function taking the value ∞ everywhere. The functions of C(z) form an algebraic field, known as the field of rational functions on the sphere.

For example, given the function

we may define f(±5) = ∞, since the denominator is zero at z = ±5, and f(∞) = 3 since f(z) → 3 as z → ∞. Using these definitions, f becomes a continuous function from the Riemann sphere to itself.

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 C. Let ζ be a complex number in one copy of C, and let ξ be a complex number in another copy of C. Identify each nonzero complex number ζ of the first C with the nonzero complex number 1/ξ of the second C. Then the map

is called the transition map between the two copies of C — the so-called charts — glueing them together. Since the transition maps are holomorphic, they define a complex manifold, called the Riemann sphere. As a complex manifold of 1 complex dimension (i.e., 2 real dimensions), this is also called a Riemann 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 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 "∞" in the ζ-chart. Symmetrically, the origin of the ζ-chart plays the role of ∞ in 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 C.

On the other hand, the uniformization theorem, a central result in the classification of Riemann surfaces, states that every simply-connected Riemann surface is biholomorphic to the complex plane, the hyperbolic plane, or the Riemann sphere. Of these, the Riemann sphere is the only one that is a closed surface (a compact surface without boundary). Hence the two-dimensional sphere admits a unique complex structure turning it into a one-dimensional complex manifold.

As the complex projective line

The Riemann sphere can also be defined as the complex projective line. The points of the complex projective line are equivalence classes established by the following relation on points from C2 \ {(0,0)}:

If for some λ ≠ 0, w = λu and z = λv, then

In this case the equivalence class is written [w, z] using projective coordinates. Given any point [w, z] in the complex projective line, one of w and z must be non-zero, say w ≠ 0. Then by the equivalence relation,

which is in a chart for the Riemann sphere manifold. [2]

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.

As a sphere

Stereographic projection of a complex number A onto a point a of the Riemann sphere RiemannSphere.png
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 R3. 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 (θ, φ) on the sphere (with θ the zenith and φ the azimuth), the projection is

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

In order to cover the unit sphere, one needs the two stereographic projections: the first will cover the whole sphere except the point (0, 0, 1) and the second except the point (0, 0, −1). Hence, one needs two complex planes, one for each projection, which can be intuitively seen as glued back-to-back at z = 0. Note that 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.

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.


A Riemann surface does not come equipped with any particular Riemannian 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 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.

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 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/K in R3 via stereographic projection. In the ζ-chart on the Riemann sphere, the metric with K = 1 is given by

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

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).

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

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, up to conformal equivalence. 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 (with any fixed radius, though radius = 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 SO(3), a continuous ("Lie") group that is topologically the 3-dimensional projective space P3.)


A Mobius transformation acting on the sphere, and on the plane by stereographic projection Mob3d-elip-opp-200.png
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 conformal map (i.e. 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

where a, b, c, and d are complex numbers such that adbc ≠ 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.

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

Thus the Möbius transformations can be described as 2 × 2 complex matrices with nonzero determinant. Since they act on projective coordinates, two matrices yield the same Möbius transformation if and only if they differ by a nonzero factor. The group of Möbius transformations is the projective linear group PGL(2, C).

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 PGL(2, C), namely PSU(2). This subgroup is isomorphic to the rotation group SO(3), which is the group of symmetries of the unit sphere in R3 (which, when restricted to the sphere, become the isometries of the sphere).


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 (see also Quantum bit and Bloch sphere). The Riemann sphere has been suggested as a relativistic model for the celestial sphere. [3] 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.

See also

Related Research Articles

Surface (topology) Two-dimensional manifold

In the part of mathematics referred to as topology, a surface is a two-dimensional manifold. Some surfaces arise as the boundaries of three-dimensional solids; for example, the sphere is the boundary of the solid ball. Other surfaces arise as graphs of functions of two variables; see the figure at right. However, surfaces can also be defined abstractly, without reference to any ambient space. For example, the Klein bottle is a surface that cannot be embedded in three-dimensional Euclidean space.

Conformal map Mathematical function which preserves angles

In mathematics, a conformal map is a function that locally preserves angles, but not necessarily lengths.

Cauchys integral formula Provides integral formulas for all derivatives of a holomorphic function

In mathematics, Cauchy's integral formula, named after Augustin-Louis Cauchy, is a central statement in complex analysis. It expresses the fact that a holomorphic function defined on a disk is completely determined by its values on the boundary of the disk, and it provides integral formulas for all derivatives of a holomorphic function. Cauchy's formula shows that, in complex analysis, "differentiation is equivalent to integration": complex differentiation, like integration, behaves well under uniform limits – a result that does not hold in real analysis.

Meromorphic function Class of mathematical function

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 Dexcept for a set of isolated points, which are poles of the function. The term comes from the Ancient Greek meros (μέρος), meaning "part".

Stereographic projection Particular mapping that projects a sphere onto a plane

In geometry, the stereographic projection is a particular mapping (function) that projects a sphere onto a plane. The projection is defined on the entire sphere, except at one point: the projection point. Where it is defined, the mapping is smooth and bijective. It is conformal, meaning that it preserves angles at which curves meet. It is neither isometric nor area-preserving: that is, it preserves neither distances nor the areas of figures.

Riemann surface One-dimensional complex manifold

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.

Complex geometry Study of complex manifolds and several complex variables

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.

Unit disk

In mathematics, the open unit disk around P, is the set of points whose distance from P is less than 1:

In mathematics, conformal geometry is the study of the set of angle-preserving (conformal) transformations on a space.

In complex analysis, Liouville's theorem, named after Joseph Liouville, states that every bounded entire function must be constant. That is, every holomorphic function for which there exists a positive number such that for all in is constant. Equivalently, non-constant holomorphic functions on have unbounded images.

In geometry and complex analysis, a Möbius transformation of the complex plane is a rational function of the form

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. The function is n-tuples of complex numbers, classically considered on the complex coordinate space .

In mathematics, a linear fractional transformation is, roughly speaking, a transformation of the form

In differential geometry and complex geometry, a complex manifold is a manifold with an atlas of charts to the open unit disk in Cn, such that the transition maps are holomorphic.

In mathematics, the Schwarzian derivative, named after the German mathematician Hermann Schwarz, is a certain operator that is invariant under all Möbius transformations. Thus, it occurs in the theory of the complex projective line, and in particular, in the theory of modular forms and hypergeometric functions. It plays an important role in the theory of univalent functions, conformal mapping and Teichmüller spaces.

In mathematics, a Paley–Wiener theorem is any theorem that relates decay properties of a function or distribution at infinity with analyticity of its Fourier transform. The theorem is named for Raymond Paley (1907–1933) and Norbert Wiener (1894–1964). The original theorems did not use the language of distributions, and instead applied to square-integrable functions. The first such theorem using distributions was due to Laurent Schwartz. These theorems heavily rely on the triangle inequality.

Schwarz lemma

In mathematics, the Schwarz lemma, named after Hermann Amandus Schwarz, is a result in complex analysis about holomorphic functions from the open unit disk to itself. The lemma is less celebrated than deeper theorems, such as the Riemann mapping theorem, which it helps to prove. It is, however, one of the simplest results capturing the rigidity of holomorphic functions.

In mathematics, a fundamental polygon can be defined for every compact Riemann surface of genus greater than 0. It encodes not only the topology of the surface through its fundamental group but also determines the Riemann surface up to conformal equivalence. By the uniformization theorem, every compact Riemann surface has simply connected universal covering surface given by exactly one of the following:

In mathematics, and specifically in potential theory, the Poisson kernel is an integral kernel, used for solving the two-dimensional Laplace equation, given Dirichlet boundary conditions on the unit disk. The kernel can be understood as the derivative of the Green's function for the Laplace equation. It is named for Siméon Poisson.

Clifford analysis, using Clifford algebras named after William Kingdon Clifford, is the study of Dirac operators, and Dirac type operators in analysis and geometry, together with their applications. Examples of Dirac type operators include, but are not limited to, the Hodge–Dirac operator, on a Riemannian manifold, the Dirac operator in euclidean space and its inverse on and their conformal equivalents on the sphere, the Laplacian in euclidean n-space and the Atiyah–Singer–Dirac operator on a spin manifold, Rarita–Schwinger/Stein–Weiss type operators, conformal Laplacians, spinorial Laplacians and Dirac operators on SpinC manifolds, systems of Dirac operators, the Paneitz operator, Dirac operators on hyperbolic space, the hyperbolic Laplacian and Weinstein equations.


  1. B. Riemann: Theorie der Abel'sche Funktionen, J. Math. (Crelle) 1857; Werke 88-144. The name is due to Neumann C :Vorlesungen über Riemanns Theorie der Abelsche Integrale, Leipzig 1865 (Teubner)
  2. William Mark Goldman (1999) Complex Hyperbolic Geometry, page 1, Clarendon Press ISBN   0-19-853793-X
  3. R. Penrose (2007). The Road to Reality . Vintage books. pp. 428–430 (§18.5). ISBN   0-679-77631-1.