Riemann sphere

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 ${\displaystyle \mathbf{C}}$. Let ${\displaystyle \zeta }$ be a complex number in one copy of ${\displaystyle \mathbf{C}}$, and let ${\displaystyle \xi }$ be a complex number in another copy of ${\displaystyle \mathbf{C}}$. Identify each nonzero complex number ${\displaystyle \zeta }$ of the first ${\displaystyle \mathbf{C}}$ with the nonzero complex number ${\displaystyle 1/\xi }$ of the second ${\displaystyle \mathbf{C}}$. Then the map

${\displaystyle f(z)=\frac{1}{z}}$

is called the transition map between the two copies of ${\displaystyle \mathbf{C}}$—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 ${\displaystyle \zeta }$ value and a ${\displaystyle \xi }$ value, and the two values are related by ${\displaystyle \zeta =1/\xi }$. The point where ${\displaystyle \xi =0}$ should then have ${\displaystyle \zeta }$-value "${\displaystyle 1/0}$"; in this sense, the origin of the ${\displaystyle \xi }$-chart plays the role of ${\displaystyle \mathrm{\infty }}$ in the ${\displaystyle \zeta }$-chart. Symmetrically, the origin of the ${\displaystyle \zeta }$-chart plays the role of ${\displaystyle \mathrm{\infty }}$ in the ${\displaystyle \xi }$-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 ${\displaystyle \mathbf{C}}$.

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 can be defined as equivalence classes of non-null vectors in the complex vector space ${\displaystyle {\mathbf{C}}^2}$: two non-null vectors ${\displaystyle (w,z)}$ and ${\displaystyle (u,v)}$ are equivalent iff ${\displaystyle (w,z)=(\lambda u,\lambda v)}$ for some non-zero coefficient ${\displaystyle \lambda \in \mathbf{C}}$.

In this case, the equivalence class is written ${\displaystyle [w,z]}$ using projective coordinates. Given any point ${\displaystyle [w,z]}$ in the complex projective line, one of ${\displaystyle w}$ and ${\displaystyle z}$ must be non-zero, say ${\displaystyle w\ne 0}$. Then by the notion of equivalence, ${\displaystyle [w,z]=\left[1,z/w\right]}$, 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 ${\displaystyle x^2+y^2+z^2=1}$ in the three-dimensional real space ${\displaystyle {\mathbf{R}}^3}$. To this end, consider the stereographic projection from the unit sphere minus the point ${\displaystyle (0,0,1)}$ onto the plane ${\displaystyle z=0,}$ which we identify with the complex plane by ${\displaystyle \zeta =x+iy}$. In Cartesian coordinates ${\displaystyle (x,y,z)}$ and spherical coordinates ${\displaystyle (\theta ,\phi )}$ on the sphere (with ${\displaystyle \theta }$ the zenith angle and ${\displaystyle \phi }$ the azimuth), the projection is

${\displaystyle \zeta =\frac{x+iy}{1-z}=\cot (\frac{1}{2}\theta )e^{i\phi }.}$

Similarly, stereographic projection from ${\displaystyle (0,0,-1)}$ onto the plane ${\displaystyle z=0,}$ identified with another copy of the complex plane by ${\displaystyle \xi =x-iy,}$ is written

${\displaystyle \xi =\frac{x-iy}{1+z}=\tan (\frac{1}{2}\theta )e^{-i\phi }.}$

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 ${\displaystyle (0,0,1)}$, and the second covers the sphere except the point ${\displaystyle (0,0,-1)}$. The two complex planes, that are the domains of these maps, are identified differently with the plane ${\displaystyle z=0}$, because an orientation-reversal is necessary to maintain consistent orientation on the sphere.

The transition maps between ${\displaystyle \zeta }$-coordinates and ${\displaystyle \xi }$-coordinates are obtained by composing one projection with the inverse of the other. They turn out to be ${\displaystyle \zeta =1/\xi }$ and ${\displaystyle \xi =1/\zeta }$, as described above. Thus the unit sphere is diffeomorphic to the Riemann sphere.

Under this diffeomorphism, the unit circle in the ${\displaystyle \zeta }$-chart, the unit circle in the ${\displaystyle \xi }$-chart, and the equator of the unit sphere are all identified. The unit disk ${\displaystyle |\zeta |<1}$ is identified with the southern hemisphere ${\displaystyle z<0}$, while the unit disk ${\displaystyle |\xi |<1}$ is identified with the northern hemisphere ${\displaystyle z>0}$.

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 ${\displaystyle \gamma }$ must have positive curvature ${\displaystyle K}$. It follows that the metric must be isometric to the sphere of radius ${\displaystyle 1/\sqrt{K}}$ in ${\displaystyle {\mathbf{R}}^3}$ via stereographic projection. In the ${\displaystyle \zeta }$-chart on the Riemann sphere, the metric with ${\displaystyle K=1}$ is given by

${\displaystyle d{s}^2=2{\gamma }_{\zeta \overline{\zeta }}d\zeta d\overline{\zeta }=\frac{4}{{\left(1+\zeta \overline{\zeta }\right)}^2}d\zeta d\overline{\zeta }={\left(\frac{2}{1+|\zeta {|}^2}\right)}^2|d\zeta {|}^2.}$

In real coordinates ${\displaystyle \zeta =u+iv}$, the formula is

${\displaystyle d{s}^2=\frac{4}{{\left(1+u^2+v^2\right)}^2}\left(d{u}^2+d{v}^2\right).}$

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). The two non-vanishing Christoffel symbols of its Levi-Civita connection are ${\displaystyle {\mathrm{\Gamma }}_{\zeta \zeta }^{\zeta }=-2\overline{\zeta }/(1+|\zeta {|}^2)}$ and its conjugate. This metric is therefore equal to its own Ricci curvature, ${\displaystyle \gamma =\mathrm{R}\mathrm{i}\mathrm{c}}$.

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 ${\displaystyle \text{SO}(3)}$. In this sense, this is by far the most symmetric metric on the sphere. (The group of all isometries, known as ${\displaystyle \text{O}(3)}$, is also 3-dimensional, but unlike ${\displaystyle \text{SO}(3)}$ is not a connected space.)

Conversely, let ${\displaystyle S}$ denote the sphere (as an abstract smooth or topological manifold). By the uniformization theorem there exists a unique complex structure on ${\displaystyle S}$ up to conformal equivalence. It follows that any metric on ${\displaystyle S}$ 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 ${\displaystyle 1}$ 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 ${\displaystyle \text{SO}(3)}$, a continuous ("Lie") group that is topologically the 3-dimensional projective space ${\displaystyle {\mathbf{P}}^3}$.)

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

${\displaystyle f(\zeta )=\frac{a\zeta +b}{c\zeta +d},}$

where ${\displaystyle a}$, ${\displaystyle b}$, ${\displaystyle c}$, and ${\displaystyle d}$ are complex numbers such that ${\displaystyle ad-bc\ne 0}$. 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

${\displaystyle [\zeta ,1]\left(\begin{array}{cc}a& c\\ b& d\end{array}\right)=[a\zeta +b,c\zeta +d]=\left(\begin{array}{cc}\frac{a\zeta +b}{c\zeta +d}& 1\end{array}\right)=[f(\zeta ),1].}$

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 ${\displaystyle \text{PGL}(2,\mathbf{C})}$.

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 ${\displaystyle \text{PGL}(2,\mathbf{C})}$, namely ${\displaystyle \text{PSU}(2)}$. This subgroup is isomorphic to the rotation group ${\displaystyle \text{SO}(3)}$, which is the group of symmetries of the unit sphere in ${\displaystyle {\mathbf{R}}^3}$ (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 ${\displaystyle f/g}$ of two holomorphic functions ${\displaystyle f}$ and ${\displaystyle g}$. As a map to the complex numbers, it is undefined wherever ${\displaystyle g}$ is zero. However, it induces a holomorphic map ${\displaystyle (f,g)}$ to the complex projective line that is well-defined even where ${\displaystyle g=0}$. 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 is often cited as a construction on which one can easily visualise generalised circles, Möbius transformations and conformal maps between connected open subsets of the extended complex plane.

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 ${\displaystyle 1/2}$, 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. 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.