Other meanings of Riemann sphere
COMPLEX ANALYSIS
The Riemann sphere is a geometric model of the extended complex plane, formed by adjoining one ideal point, denoted ∞, to the complex numbers. It converts many statements about behavior at infinity into ordinary statements about a compact surface.
The Riemann sphere is the complex plane together with a single point at infinity, written ℂ̂ = ℂ ∪ {∞}.1 A standard construction places the complex plane on a sphere by stereographic projection: each complex number corresponds to a point on the sphere, while the projection point represents ∞. The resulting space is topologically a two-dimensional sphere and, more specifically, a one-dimensional complex manifold.
Unlike the real line, which has two directions toward infinity, the complex plane has one coherent end in the topological sense. Every path whose modulus tends to infinity approaches the same added point. Neighborhoods of ∞ are therefore sets containing all sufficiently large complex numbers, together with ∞ itself.
The sphere remains a complex plane locally even at infinity because the coordinate w = 1/z provides a chart there. Away from z = 0, inversion exchanges the ordinary coordinate z with w; as z tends to infinity, w tends to 0. This makes ∞ an ordinary point in the coordinate w, rather than a singular location in the surface itself.
The two charts z and w cover the sphere, with transition function w = 1/z on their overlap. That transition is holomorphic wherever both coordinates are defined, which gives the Riemann sphere its complex-manifold structure. Its compactness is central: every sequence has a convergent subsequence after allowing convergence to ∞, and every holomorphic function from the sphere to the complex plane is constant.2
Meromorphic functions on the complex plane become holomorphic maps to the Riemann sphere when poles are assigned the value ∞.1 A rational function therefore defines a globally meaningful map from the sphere to itself: at a pole it takes the value ∞, and at ∞ its value is determined by the leading terms. This viewpoint unifies zeros and poles as preimages of 0 and ∞.
The automorphisms of the Riemann sphere are exactly the Möbius transformations f(z) = (az + b)/(cz + d), where ad − bc ≠ 0; they act continuously and holomorphically on the extended plane. Such transformations include translations, rotations, dilations, inversion, and their compositions. They preserve generalized circles: ordinary circles and straight lines are treated as circles passing through ∞.
The Riemann sphere is not merely a picture of infinity; it is the basic compact model behind several classification results in complex analysis.2 The point ∞ can be an ordinary value, a pole, or an essential singularity depending on the function under study. For example, a polynomial has a pole at ∞, while ez has an essential singularity there; examining f(1/w) near w = 0 makes these distinctions precise.
The sphere also carries a natural orientation and conformal geometry, so angles are preserved by stereographic projection except at the projection point, where the coordinate description changes. In algebraic geometry, the complex projective line is naturally identified with the Riemann sphere, written ℂℙ1. This identification connects elementary complex analysis with projective geometry, algebraic curves, and the classification of compact Riemann surfaces of genus zero.3
The notation ℂ̂ is commonly used for the extended complex plane; conventions for assigning values at poles and at infinity are understood through limits in the local coordinate w = 1/z.
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