The Riemann Sphere

# The Riemann Sphere

 Definition: The Riemann Sphere denoted $\mathbb{C}_{\infty} = \mathbb{C} \cup \{ \infty \}$ is the topological space adjoining the single point $\infty$ to $\mathbb{C}$.

We can readily define a very simple two chart atlas on $\mathbb{C}_{\infty}$, call it $\mathcal M = \{ (U_0, \phi_0), (U_{\infty}, \phi_{\infty}) \}$ where:

(1)
\begin{align} \quad U_0 &= \mathbb{C} \\ \quad \phi_0 &: U_0 \to \mathbb{C} \: \mathrm{is \: defined \: for \: all \:} z \in \mathbb{C} \: \mathrm{by} \: \phi_0(z) = z \\ \quad U_{\infty} &= \mathbb{C}_{\infty} \setminus \{ 0 \} \\ \quad \phi_{\infty} &: U_{\infty} \to \mathbb{C} \: \mathrm{is \: defined \: for \: all \:} z \in \mathbb{C}_{\infty} \setminus \{ 0 \} \: \mathrm{by} \: \phi_{\infty}(z) = \left\{\begin{matrix} \frac{1}{z} & \mathrm{if} \: z \neq \infty \\ 0 & \mathrm{if} \: z = \infty \end{matrix}\right. \end{align}

Clearly $\phi_0$ and $\phi_{\infty}$ are homeomorphisms onto their images.

Now, the Riemann sphere can be modelled via stereographc projection. Consider the unit sphere centered at the origin in $\mathbb{R}^3$ with equation $x_1^2 + x_2^2 + x_3^2 = 1$ with north pole $(x_1, x_2, x_3) = (0, 0, 1)$ and identify the $x_1x_2$-plane embedded in $\mathbb{R}^3$ with the complex plane $\mathbb{C}$. The north pole of the sphere has coordinates $(x, y, z) = (1, 1, 1)$. For every complex number $z = x + iy$ there is a corresponding point $(x_1, x_2, x_3)$ on the sphere, and this point is given by the equations:

(2)
\begin{align} \quad x_1 = \frac{2x}{1 + |z|^2} \quad \quad x_2 = \frac{2y}{1 + |z|^2} \quad \quad x_3 = \frac{|z|^2 - 1}{1 + |z|^2} \end{align}