Riemann Sphere - Automorphisms

Automorphisms

The study of any mathematical object is aided by an understanding of its group of automorphisms, meaning the maps from the object to itself that preserve the essential structure of the object. In the case of the Riemann sphere, an automorphism is an invertible biholomorphic map from the Riemann sphere to itself. It turns out that the only such maps are the Möbius transformations. These are functions of the form

where a, b, c, and d are complex numbers such that . Examples of Möbius transformations include dilations, rotations, translations, and complex inversion. In fact, any Möbius transformation can be written as a composition of these.

The Möbius transformations are profitably viewed as transformations on the complex projective line. In projective coordinates, the transformation f' can be written

Thus the Möbius transformations can be described as 2 × 2 complex matrices with nonzero determinant; two matrices yield the same Möbius transformation if and only if they differ by a nonzero factor. Thus the Möbius transformations exactly correspond to the projective linear transformations PGL(2, C).

If one endows the Riemann sphere with the Fubini–Study metric, then not all Möbius transformations are isometries; for example, the dilations and translations are not. The isometries form a proper subgroup of PGL(2, C), namely PSU(2). This subgroup is isomorphic to the rotation group SO(3), which is the group of symmetries of the unit sphere in R3 (which, when restricted to the sphere, become the isometries of the sphere).