page_title: Riemann sphere
section_title: Automorphisms
hierarchical_section_title: Riemann sphere / Automorphisms
context_page_description: In mathematics, the Riemann sphere, named after Bernhard Riemann, is a model of the extended complex plane, the complex plane plus a point at infinity. This extended plane represents the extended complex numbers, that is, the complex numbers plus a value ∞ for infinity. With the Riemann model, the point "∞" is near to very large numbers, just as the point "0" is near to very small numbers.
  The extended complex numbers are useful in complex analysis because they allow for division by zero in some circumstances, in a way that makes expressions such as well-behaved. For example, any rational function on the complex plane can be extended to a holomorphic function on the Riemann sphere, with the poles of the rational function mapping to infinity. More generally, any meromorphic function can be thought of as a holomorphic function whose codomain is the Riemann sphere.
  In geometry, the Riemann sphere is the prototypical example of a Riemann surface, and is one of the simplest complex manifolds. In projective geometry, the sphere can be thought of as the complex projective line P¹, the projective space of all complex lines in C².
context_section_description: 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 ad − bc ≠ 0. 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 homographies 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. Since they act on projective coordinates, two matrices yield the same Möbius transformation if and only if they differ by a nonzero factor. The group of Möbius transformations is the projective linear group 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).
caption: A Möbius transformation acting on the sphere, and on the plane by stereographic projection
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