Editing Möbius transformation (section)

== Overview ==
Möbius transformations are defined on the [[Riemann sphere|extended complex plane]] <math>\widehat{\Complex} = \Complex \cup \{\infty\}</math> (i.e., the [[complex plane]] augmented by the [[point at infinity]]).

[[Stereographic projection]] identifies <math>\widehat{\Complex}</math> with a sphere, which is then called the [[Riemann sphere]]; alternatively, <math>\widehat{\Complex}</math> can be thought of as the complex [[projective line]] <math>\Complex\mathbb{P}^1</math>.  The Möbius transformations are exactly the [[bijective]] [[conformal map|conformal]] maps from the Riemann sphere to itself, i.e., the [[automorphism group|automorphisms]] of the Riemann sphere as a [[complex manifold]]; alternatively, they are the automorphisms of <math>\Complex\mathbb{P}^1</math> as an algebraic variety.  Therefore, the set of all Möbius transformations forms a [[group (mathematics)|group]] under [[function composition|composition]].  This group is called the Möbius group, and is sometimes denoted <math>\operatorname{Aut}(\widehat{\Complex})</math>.

The Möbius group is [[group isomorphism|isomorphic]] to the group of orientation-preserving [[isometry|isometries]] of [[hyperbolic space|hyperbolic 3-space]] and therefore plays an important role when studying [[hyperbolic 3-manifold]]s.

In [[physics]], the [[identity component]] of the [[Lorentz group]] acts on the [[celestial sphere]] in the same way that the Möbius group acts on the Riemann sphere. In fact, these two groups are isomorphic. An observer who accelerates to relativistic velocities will see the pattern of constellations as seen near the Earth continuously transform according to infinitesimal Möbius transformations. This observation is often taken as the starting point of [[twistor theory]].

Certain [[subgroup]]s of the Möbius group form the automorphism groups of the other [[simply-connected]] Riemann surfaces (the [[complex plane]] and the [[Hyperbolic space|hyperbolic plane]]). As such, Möbius transformations play an important role in the theory of [[Riemann surface]]s. The [[fundamental group]] of every Riemann surface is a [[discrete subgroup]] of the Möbius group (see [[Fuchsian group]] and [[Kleinian group]]). A particularly important discrete subgroup of the Möbius group is the [[modular group]]; it is central to the theory of many [[fractal]]s, [[modular form]]s, [[elliptic curve]]s and [[Pellian equation]]s.

Möbius transformations can be more generally defined in spaces of dimension {{math|''n'' > 2}} as the bijective conformal orientation-preserving maps from the [[n-sphere|{{nowrap|{{mvar|n}}-sphere}}]] to the {{mvar|n}}-sphere. Such a transformation is the most general form of conformal mapping of a domain. According to [[Liouville's theorem (conformal mappings)|Liouville's theorem]] a Möbius transformation can be expressed as a composition of translations, [[Similarity (geometry)|similarities]], orthogonal transformations and inversions.