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

is called the transition map between the two copies of C {\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 ζ {\displaystyle \zeta } value and a ξ {\displaystyle \xi } value, and the two values are related by ζ = 1 / ξ {\displaystyle \zeta =1/\xi }. The point where ξ = 0 {\displaystyle \xi =0} should then have ζ {\displaystyle \zeta } -value " 1 / 0 {\displaystyle 1/0} "; in this sense, the origin of the ξ {\displaystyle \xi } -chart plays the role of ∞ {\displaystyle \infty } in the ζ {\displaystyle \zeta } -chart. Symmetrically, the origin of the ζ {\displaystyle \zeta } -chart plays the role of ∞ {\displaystyle \infty } in the ξ {\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 C {\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.
oulipo:avoid_letter[t], detectable_content:postscript