Riemann Sphere - As The Complex Projective Line

As The Complex Projective Line

The Riemann sphere can also be defined as the complex projective line. This is the subset of C2 consisting of all pairs (α, β) of complex numbers, not both zero, modulo the equivalence relation

for all nonzero complex numbers λ. The complex plane C, with coordinate ζ, can be mapped into the complex projective line by

Another copy of C with coordinate ξ can be mapped in by

These two complex charts cover the projective line. For nonzero ξ the identifications

demonstrate that the transition maps are ζ = 1/ξ and ξ = 1/ζ, as above.

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.