Riemann sphere
Definition
As a Riemann surface
The Riemann sphere is the topological space (the unit sphere in 3-space) with the following atlas of charts:
- The stereographic projection that maps the complement of the north pole to
- The stereographic projection that maps the complement of the south pole to , composed with a reflection about the -axis in
The transition map between these two charts is given by:
Other descriptions
The Riemann sphere is viewed in many of the following ways:
- It is the one-point compactification of , the one additional point being a point at infinity, denoted . With respect to the stereographic projection, the point at infinity is identified with the north pole.
- It is the complex projective line: the set of complex lines through the origin in .