Model of the extended complex plane plus a point at infinity
In mathematics, the Riemann sphere, named after Bernhard Riemann,
 is a model of the extended complex plane (also called the closed complex plane): the complex plane plus one point at infinity. This extended plane represents the extended complex numbers, that is, the complex numbers plus a value formula_1 for infinity. With the Riemann model, the point formula_1 is near to very large numbers, just as the point formula_3 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 formula_4 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 is an example of a complex projective space and can be thought of as the complex projective line formula_5, the projective space of all complex lines in formula_6. 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.
Extended complex numbers.
The extended complex numbers consist of the complex numbers formula_7 together with formula_1. The set of extended complex numbers may be written as formula_9, and is often denoted by adding some decoration to the letter formula_7, such as
formula_11
The notation formula_12 has also seen use, but as this notation is also used for the punctured plane formula_13, it can lead to ambiguity.
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 formula_14,
formula_15
for any complex number formula_16, and multiplication may be defined by
formula_17
for all nonzero complex numbers formula_16, with formula_19. Note that formula_20 and formula_21 are left undefined. Unlike the complex numbers, the extended complex numbers do not form a field, since formula_1 does not have an additive nor multiplicative inverse. Nonetheless, it is customary to define division on formula_9 by
formula_24
for all nonzero complex numbers formula_16 with formula_26 and formula_27. The quotients formula_28 and formula_29 are left undefined.
Rational functions.
Any rational function formula_30 (in other words, formula_31 is the ratio of polynomial functions formula_32 and formula_33 of formula_16 with complex coefficients, such that formula_32 and formula_33 have no common factor) can be extended to a continuous function on the Riemann sphere. Specifically, if formula_37 is a complex number such that the denominator formula_38 is zero but the numerator formula_39 is nonzero, then formula_40 can be defined as formula_1. Moreover, formula_42 can be defined as the limit of formula_31 as formula_44, which may be finite or infinite.
The set of complex rational functions—whose mathematical symbol is formula_45—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 formula_1 everywhere. The functions of formula_45 form an algebraic field, known as "the field of rational functions on the sphere".
For example, given the function
formula_48
we may define formula_49, since the denominator is zero at formula_50, and formula_51 since formula_52 as formula_44. Using these definitions, formula_54 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 formula_7. Let formula_56 be a complex number in one copy of formula_57, and let formula_58 be a complex number in another copy of formula_59. Identify each nonzero complex number formula_56 of the first formula_57 with the nonzero complex number formula_62 of the second formula_59. Then the map
formula_64
is called the transition map between the two copies of formula_59—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 formula_56 value and a formula_67 value, and the two values are related by formula_68. The point where formula_69 should then have formula_56-value "formula_71"; in this sense, the origin of the formula_58-chart plays the role of formula_73 in the formula_56-chart. Symmetrically, the origin of the formula_56-chart plays the role of formula_73 in the formula_58-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 formula_7.
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 can be defined as equivalence classes of "non-null vectors" in the complex vector space formula_6: two non-null vectors formula_80 and formula_81 are equivalent iff formula_82 for some non-zero coefficient formula_83.
In this case, the equivalence class is written formula_84 using projective coordinates. Given any point formula_85 in the complex projective line, one of formula_86 and formula_16 must be non-zero, say formula_88. Then by the notion of equivalence, formula_89, which is in a chart for the Riemann sphere manifold.
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.
The Riemann sphere can be visualized as the unit sphere formula_90 in the three-dimensional real space formula_91. To this end, consider the stereographic projection from the unit sphere minus the point formula_92 onto the plane formula_93 which we identify with the complex plane by formula_94. In Cartesian coordinates formula_95 and spherical coordinates formula_96 on the sphere (with formula_97 the zenith and formula_98 the azimuth), the projection is
formula_99
Similarly, stereographic projection from formula_100 onto the plane formula_93 identified with another copy of the complex plane by formula_102 is written
formula_103
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 formula_104, and the second covers the sphere except the point formula_105. The two complex planes, that are the domains of these maps, are identified differently with the plane formula_106, because an orientation-reversal is necessary to maintain consistent orientation on the sphere.
The transition maps between formula_107-coordinates and formula_108-coordinates are obtained by composing one projection with the inverse of the other. They turn out to be formula_109 and formula_110, as described above. Thus the unit sphere is diffeomorphic to the Riemann sphere.
Under this diffeomorphism, the unit circle in the formula_107-chart, the unit circle in the formula_108-chart, and the equator of the unit sphere are all identified. The unit disk formula_113 is identified with the southern hemisphere formula_114, while the unit disk formula_115 is identified with the northern hemisphere formula_116.
Metric.
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 formula_117. It follows that the metric must be isometric to the sphere of radius formula_118 in formula_119 via stereographic projection. In the formula_120-chart on the Riemann sphere, the metric with formula_121 is given by
formula_122
In real coordinates formula_123, the formula is
formula_124
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 formula_125. In this sense, this is by far the most symmetric metric on the sphere. (The group of all isometries, known as formula_126, is also 3-dimensional, but unlike formula_125 is not a connected space.)
Conversely, let formula_128 denote the sphere (as an abstract smooth or topological manifold). By the uniformization theorem there exists a unique complex structure on formula_128 up to conformal equivalence. It follows that any metric on formula_128 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 formula_131 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 formula_132, a continuous ("Lie") group that is topologically the 3-dimensional projective space formula_133.)
Automorphisms.
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
formula_134
where formula_135, formula_136, formula_137, and formula_138 are complex numbers such that formula_139. 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
formula_140
Thus the Möbius transformations can be described as two-by-two 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 formula_141.
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 formula_141, namely formula_143. This subgroup is isomorphic to the rotation group formula_144, which is the group of symmetries of the unit sphere in formula_145 (which, when restricted to the sphere, become the isometries of the sphere).
Applications.
In complex analysis, a meromorphic function on the complex plane (or on any Riemann surface, for that matter) is a ratio formula_146 of two holomorphic functions formula_147 and formula_148. As a map to the