Riemann sphere

Definition

As a Riemann surface

The Riemann sphere is the topological space $S^{2}\subset \mathbb {R} ^{3}$ (the unit sphere in 3-space) with the following atlas of charts:

• The stereographic projection that maps the complement of the north pole to $\mathbb {C}$
• The stereographic projection that maps the complement of the south pole to $\mathbb {C}$, composed with a reflection about the $x$-axis in $\mathbb {C}$

The transition map between these two charts is given by:

$z\mapsto {\frac {1}{z}}$

Other descriptions

The Riemann sphere is viewed in many of the following ways:

• It is the one-point compactification of $\mathbb {C}$, the one additional point being a point at infinity, denoted $\infty$. 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 $\mathbb {C} ^{2}$.