Stereographic projection identifies a sphere minus one pole with the plane and supplies the standard charts of the Riemann sphere.
On the unit sphere, use away from the north pole and away from the south pole. Their overlap relation is a holomorphic map, giving a holomorphic atlas for the Riemann sphere. The opposite sign of in the second manifold chart ensures a holomorphic rather than antiholomorphic transition.
Under the stereographic projection , rotations about the third axis have representatives , and rotations about the second axis have representativesThese belong to the special unitary group . Euler-angle generation of the special orthogonal group shows that every sphere rotation becomes a Möbius transformation represented in . Representatives and induce the same transformation.
Let a finite group of conformal automorphisms of the Riemann sphere fix . The derivative representation at embeds into , so is cyclic. Averaging a local coordinate linearizes the action to , and the quotient has coordinate .
For a finite conformal group action, choose a disc around each point that meets only its stabilizer translates. The local cyclic quotient chart , where is the stabilizer order, gives the orbit space a Riemann-surface structure and makes the quotient map holomorphic.
For and , where , the rational functionis invariant under . Equality of two values factors as for and , so every fibre is exactly one dihedral orbit.
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The Riemann sphere is a model for visualizing complex numbers and their geometric properties in a compact form. It is named after the German mathematician Bernhard Riemann. The Riemann sphere is essentially a way of extending the complex plane by adding a point at infinity, allowing for a more complete understanding of complex functions, including those that have poles or essential singularities.