Riemann Sphere - As A Sphere

As A Sphere

The Riemann sphere can be visualized as the unit sphere x2 + y2 + z2 = 1 in the three-dimensional real space R3. To this end, consider the stereographic projection from the unit sphere minus the point (0, 0, 1) onto the plane z = 0, which we identify with the complex plane by ζ = x + iy. In Cartesian coordinates (x, y, z) and spherical coordinates (φ, θ) on the sphere (with φ the zenith and θ the azimuth), the projection is

Similarly, stereographic projection from (0, 0, −1) onto the plane z = 0, identified with another copy of the complex plane by ξ = x - i y, is written

In order to cover the unit sphere, one needs the two stereographic projections: the first will cover the whole sphere except the point (0,0,1) and the second except the point (0,0,-1). Hence, one needs two complex planes, one for each projection, which can be intuitively seen as glued back-to-back at z=0. Note that the two complex planes are identified differently with the plane z = 0. An orientation-reversal is necessary to maintain consistent orientation on the sphere, and in particular complex conjugation causes the transition maps to be holomorphic.

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

Under this diffeomorphism, the unit circle in the ζ-chart, the unit circle in the ξ-chart, and the equator of the unit sphere are all identified. The unit disk |ζ| < 1 is identified with the southern hemisphere z < 0, while the unit disk |ξ| < 1 is identified with the northern hemisphere z > 0.