Riemann sphere

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A loxodromic representation of the Riemann sphere.
A loxodromic representation of the Riemann sphere.

In mathematics, the Riemann sphere, named after Bernhard Riemann, is the unique way of viewing the extended complex plane (the complex plane plus a point at infinity) so that it looks exactly the same at the point infinity as at any complex number. The main application is to deal with extended complex functions (which may be defined at the point infinity and/or take the value infinity, in addition to complex numbers) in the same way at the point infinity as at any complex number, specifically with respect to continuity and differentiability.

From the geometrical view of the plane that deals with points, lines, circles and angles but not distances, the Riemann sphere is created by adding a point at infinity through which all lines cross, with parallel lines being tangent there and all other lines crossing at the same angle as they do at an existing point. This geometry is realized as a 2-dimensional sphere formed from the extended complex plane using the stereographic projection, where lines in the complex plane become circles through infinity. Angles in the Riemann sphere are identical to the corresponding angles in the complex plane (and the same is true at infinity with the natural choice of the angle between two lines at infinity).

Topologically, the Riemann sphere is the one-point compactification of the complex plane. This gives it the topology of a 2-dimensional sphere, preserving the topology of the complex plane. The Riemann sphere can be conveniently identified with a geometrical 2-dimensional sphere, in which lines become circles through infinity.

The 2-sphere admits a unique complex structure turning it into a Riemann surface (i.e. a 1-dimensional complex manifold). The Riemann sphere can be characterized as the unique simply-connected, compact Riemann surface, and may be taken to have the complex plane as a complex sub-manifold.

In all of these viewpoints, the point at infinity acquires an identical role to any point in the complex plane.

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[edit] Geometric introduction

Define \widehat{\mathbb{C}} = \mathbb{C}\cup\{\infty\} (i.e. the extended complex plane: the complex numbers joined with the point at infinity). The Riemann sphere is based on the transformation from \widehat{\mathbb{C}} to \widehat{\mathbb{C}} in the form

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

where

w, z \in \widehat{\mathbb{C}} and \frac{1}{0} = \infty.

We visualize the Riemann sphere as a sphere in 3-space, i.e. in \mathbb{R}^3. Every point on the sphere has both a z value and w value, related by the above transformation. That is, f(z) transforms the sphere onto itself.

[edit] Stereographic projection

The one to-one correspondence between a sphere (represented by a circle) and the extended complex plane (represented by a line).
The one to-one correspondence between a sphere (represented by a circle) and the extended complex plane (represented by a line).

To establish the correspondence between points in the extended complex plane and the Riemann sphere, we first place the z plane tangent to the sphere's north pole. We then use stereographic projection from the south pole of the sphere. This is done by drawing a line from the south pole that intersects both the sphere and the complex plane; a unique, one-to-one correspondence is then established between points on the complex plane and points on the Riemann sphere.

In order to complete this one-to-one correspondence for the extended complex plane, we define the south pole to be z = \infty. Note that the north pole is z = 0, following from the above logic.

The correspondence between the w plane and the Riemann sphere is done in much the same way, simply "upside down." That is, the w plane is tangent to the south pole and oriented oppositely to the z plane, such that w = 1,i, − 1, − i matches to z = 1, − i, − 1,i. We then perform the stereographic projection from the north pole, and similarly define the north pole to be w = \infty. Now, every point on the sphere has both a z and w coordinate, related by the transformation above.

The figure to the right shows two-dimensional representation of stereographic projection from the north pole. Although it is a rough analogue of the projection onto the w plane, it is not properly specialized to the Riemann sphere.

[edit] Alternate stereographic projection

An alternate version of the stereographic projection places the planes at the equator, but preserves their opposite orientation. Thus, the planes are not geometrically distinct.

This version is less popular in mathematical developments of the Riemann sphere, but seems more popular in physics. For example, it is favored by Roger Penrose in his development of twistor theory (as shown in this paper).

[edit] Geometric features of note

Stereographic projection maps all lines and circles in the complex plane to circles on the Riemann sphere. The reason that lines are mapped to circles is that a line with infinite length can simply be thought of as a circle that passes through the point at infinity.

[edit] Möbius transformations

Main article: Möbius transformation

Möbius transformations, which send \widehat{\mathbb{C}} to \widehat{\mathbb{C}}, are the automorphisms of the Riemann sphere (i.e. the conformal bijections). They are in the form

t = f(z) = \frac{az + b}{cz + d},

where t, z \in \widehat{\mathbb{C}}, a, b, c, d \in \mathbb{C}, and ad - bc \neq 0. They map the Riemann sphere to itself, preserving angles and orientation.

This may be seen directly because they may be expressed as a composition of maps of the form

z \rightarrow r z\,
z \rightarrow z e^{i\theta}\,
z \rightarrow z + z_0\,
z \rightarrow \frac{1}{z}\,

(where r and θ are real numbers and z0 is a complex number).

These are respectively, elementary dilations, rotations, translations and complex inversion (a composition of an inversion in the unit circle and a reflection in the real line), each of which is conformal on the complex plane. Using the map

z\rightarrow \frac{1}{z}\;

allows us to check that this is also true at infinity. Conversely, every everywhere-conformal bijection of the Riemann sphere is a Möbius transformation.

[edit] Complex structure

The complex manifold structure on the Riemann sphere is specified by an atlas with two charts as follows

f:\widehat{\mathbb{C}}\setminus\{\infty\} \to \mathbb{C},\ f(z)=z
g:\widehat{\mathbb{C}}\setminus\{0\} \to \mathbb{C},\ g(z)=\frac{1}{z}\mbox{ and }g(\infty) = 0.

The overlap of these two charts is all points except 0 and ∞. On this overlap the transition function is given by z → 1/z, which is clearly holomorphic and so defines a complex structure.

The Riemann sphere has the same topology as S2, that is, the sphere of radius 1 centered at the origin in the Euclidean space R3. A homeomorphism between them is given by the stereographic projection tangent to the South Pole onto the complex plane. Labeling the points in S2 by (x1, x2, x3) where x_1^2 + x_2^2 + x_3^2 = 1, the homeomorphism is

(x_1, x_2, x_3)\to \frac{x_1-i x_2}{1-x_3}.

This maps the South Pole to the origin of the complex plane and the North Pole to ∞.

In terms of standard spherical coordinates (θ, φ), this map can be given as

(\theta, \phi)\to e^{-i\phi}\cot\frac{\theta}{2}.

One can also use the stereographic projection tangent to the North Pole, which will map the North Pole to the origin and the South Pole to ∞. The formula is

(x_1, x_2, x_3) \to \frac{x_1+i x_2}{1+x_3}

or, in spherical coordinates

(\theta, \phi)\to e^{i\phi}\tan\frac{\theta}{2}.

[edit] The complex projective line

The Riemann sphere can also be realized as the complex projective line, CP1. Explicitly, the isomorphism is given by

[z_1 : z_2]\leftrightarrow z_1/z_2

where [z1 : z2] are homogeneous coordinates on CP1. Note that the complex plane sits inside the projective line as the subset

\{[z,1] : z \in \mathbb C\}

while the point at infinity is given in homogeneous coordinates by [1 : 0].

[edit] Properties

In the category of Riemann surfaces, the automorphism group of the Riemann sphere is the group of Möbius transformations. These are just the projective linear transformations PGL2 C on CP1. When the sphere is given the round metric the isometry group is the subgroup PSU2 C (which is isomorphic to rotation group SO(3)).

The Riemann sphere is one of three simply-connected Riemann surfaces, the other two being the complex plane and the hyperbolic plane. This statement, known as the uniformization theorem, is important to the classification of Riemann surfaces.

[edit] See also

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