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 $\mathbb {C} ^{2}$.

Automorphism group

Description of the automorphism group

Further information: fractional linear transformation,conformal automorphism of Riemann sphere equals fractional linear transformation

The conformal automorphisms of the Riemann sphere are the fractional linear transformations. Viewing the Riemann sphere as $\mathbb {C}$ with a point at infinity, the fractional linear transformations are maps that can be described as:

$z\mapsto {\frac {az+b}{cz+d}}$

where $ad-bc$ is nonzero. The fractional linear transformation is often represented by the matrix:

${\begin{pmatrix}a&b\\c&d\end{pmatrix}}$

and the composition of these transformations corresponds to multiplication of matrices. The fractional linear transformations also have an interpretation in terms of the Riemann sphere as the complex projective line.

The group of fractional linear transformations is the projective general linear group $PGL(2,\mathbb {C} )$.

The proof that this is the full automorphism group relies on two facts: it acts transitively, and it contains the isotropy subgroup at $\infty$. Template:Proofat

Transitivity of the automorphism group

Further information: Conformal automorphism group is 3-regular on Riemann sphere

The conformal automorphisms of the Riemann sphere act transitively on it. In fact, more is true. Given any three distinct points $z_{1},z_{2},z_{3}$, and any three distinct points $w_{1},w_{2},w_{3}$, there is a unique fractional linear transformation mapping $z_{1}$ to $w_{1}$, $z_{2}$ to $w_{2}$, and $z_{3}$ to $w_{3}$. The uniqueness follows from the fact that any fractional linear transformation with three fixed points gives a quadratic equation with three zeros, while the existence follows from a direct argument.

Isomorphism

Further information: Genus zero Riemann surface is conformally equivalent to Riemann sphere

The Riemann sphere is the only Riemann surface, up to conformal equivalence, that is topologically a sphere. In other words, any topological sphere is conformally equivalent to the Riemann sphere. Equivalently, any compact simply connected Riemann surface, or any genus zero Riemann surface, is conformally equivalent to the Riemann sphere.