Riemann sphere - Knowlpedia

The Riemann sphere is the one-point compactification

\widehat{\mathbb C}=\mathbb C\cup\{\infty\}

equipped with the complex atlas whose coordinate near finite points is $z$ and whose coordinate near $\infty$ is $w=1/z$ . It is a compact Riemann surface analytically isomorphic to the complex projective line $\mathbb P^1(\mathbb C)$ .

Projective description

The point $[z_0:z_1]\in\mathbb P^1(\mathbb C)$ corresponds to $z=z_0/z_1$ when $z_1\ne0$ , while $[1:0]$ corresponds to $\infty$ . The two affine charts are related by $w=1/z$ . This is the analytic form of the projective line, not a replacement for its scheme-theoretic structure.

Topological sphere

Stereographic projection identifies $\widehat{\mathbb C}$ homeomorphically, indeed conformally, with the round sphere $S^2$ . Neighborhoods of $\infty$ correspond to complements of compact subsets of $\mathbb C$ , realizing the one-point compactification topology.

Functions and symmetry

Holomorphic automorphisms of the sphere are exactly Möbius transformations. A meromorphic function on a plane domain is equivalently a holomorphic map into $\widehat{\mathbb C}$ , with poles sent to $\infty$ .

References

Otto Forster, Lectures on Riemann Surfaces, Springer, 1981. Publisher record. Relevant: Chapter 1, §§2 and 8.