Extended complex numbersThe 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 operationsAddition of complex numbers may be extended by defining, for ''z'' ∈ C, : for any complex number ''z'', and multiplication may be defined by : for all nonzero complex numbers ''z'', with ∞ × ∞ = ∞. Note that ∞ – ∞ and 0 × ∞ are left Indeterminate form, undefined. Unlike the complex numbers, the extended complex numbers do not form a field (mathematics), field, since ∞ does not have a multiplicative inverse. Nonetheless, it is customary to define division (mathematics), division on C ∪ by : for all nonzero complex numbers ''z'', with = ∞ and = 0. The quotients and are left undefined.
Rational functionsAny rational function (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 ''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 ∞. Moreover, ''f''(∞) can be defined as the Limit of a function, limit of ''f''(''z'') as , 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 , since the denominator is zero at , and since as . Using these definitions, ''f'' becomes a continuous function from the Riemann sphere to itself.
As a complex manifoldAs 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 of the second C. Then the map : is called the transition map between the two copies of C — the so-called Chart (topology), charts — glueing them together. Since the transition maps are holomorphic function, 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 . The point where should then have ''ζ''-value ""; 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. Topology, Topologically, the resulting space is the Compactification (mathematics), 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 manifold, complex structure, so that around every point on the sphere there is a neighborhood that can be Biholomorphism, 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 space, hyperbolic plane, or the Riemann sphere. Of these, the Riemann sphere is the only one that is a Closed manifold, closed surface (a compact space, compact surface without Manifold with boundary, boundary). Hence the two-dimensional sphere admits a unique complex structure turning it into a one-dimensional complex manifold.
As the complex projective lineThe 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 \ : :If for some λ ≠ 0, ''w'' = λ''u'' and ''z'' = λ''v'', 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. 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 sphereThe Riemann sphere can be visualized as the unit sphere ''x''2 + ''y''2 + ''z''2 = 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 . In Cartesian coordinates and spherical coordinates on the sphere (with ''θ'' the zenith and ''φ'' the azimuth), the projection is : Similarly, stereographic projection from onto the plane , identified with another copy of the complex plane by , 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 and the second except the point . Hence, one needs two complex planes, one for each projection, which can be intuitively seen as glued back-to-back at . Note that the two complex planes are identified differently with the plane . An Orientation (mathematics), 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 and , as described above. Thus the unit sphere is Diffeomorphism, 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 is identified with the southern hemisphere , while the unit disk is identified with the northern hemisphere .
MetricA 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 Gaussian curvature, curvature ''K''. It follows that the metric must be Isometry (Riemannian geometry), isometric to the sphere of radius in R3 via stereographic projection. In the ''ζ''-chart on the Riemann sphere, the metric with is given by : In real coordinates , 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 Differentiable manifold, 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 Metric tensor#The round metric on a sphere, 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 rotation group SO(3), SO(3), a continuous ("Lie") group that is topologically the 3-dimensional projective space P3.)
AutomorphismsThe study of any mathematical object is aided by an understanding of its Group (mathematics), 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 . Examples of Möbius transformations include Scaling (geometry), dilations, rotations, translation (mathematics), translations, and complex inversion. In fact, any Möbius transformation can be written as a composition of these. The Möbius transformations are homography, homographies on the complex projective line. In projective coordinates, the transformation ''f'' can be written : Thus the Möbius transformations can be described as 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 (mathematics), group of Möbius transformations is the projective linear group . 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 , 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).
ApplicationsIn complex analysis, a meromorphic function on the complex plane (or on any Riemann surface, for that matter) is a ratio 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 to the complex projective line that is well-defined even where . 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 photon polarization, polarization states, spin (physics), spin states of massive Subatomic particle, particles of spin , and 2-state particles in general (see also Quantum bit and Bloch sphere). The Riemann sphere has been suggested as a General relativity, 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.
See also*Conformal geometry *Cross-ratio *Dessin d'enfant *Directed infinity *Hopf bundle *Möbius plane *Projectively extended real line
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