Editing Möbius transformation (section)

== Higher dimensions ==
In higher dimensions, a '''Möbius transformation''' is a [[homeomorphism]] of {{tmath|1= \overline{\mathbb R^n} }}, the [[one-point compactification]] of {{tmath|1= \mathbb R^n }}, which is a finite composition of [[Inversion in a sphere|inversions in spheres]] and [[Reflection (mathematics)|reflections]] in [[hyperplanes]].<ref>Iwaniec, Tadeusz and Martin, Gaven, The Liouville theorem, Analysis and topology, 339–361, World Sci. Publ., River Edge, NJ, 1998</ref> [[Liouville's theorem (conformal mappings)|Liouville's theorem in conformal geometry]] states that in dimension at least three, all [[Conformal map|conformal]] transformations are Möbius transformations. Every Möbius transformation can be put in the form
<math display="block">f(x) = b + \frac{\alpha A(x - a)}{|x - a|^\varepsilon} ,</math>
where <math>a,b\in \mathbb R^n</math>, <math>\alpha\in\mathbb R</math>, <math>A</math> is an [[orthogonal matrix]], and <math>\varepsilon</math> is 0 or 2. The group of Möbius transformations is also called the '''Möbius group'''.<ref>J.B. Wilker (1981) "Inversive Geometry", {{mr|id=0661793}}</ref>

The orientation-preserving Möbius transformations form the connected component of the identity in the Möbius group.  In dimension {{nowrap|1=''n'' = 2}}, the orientation-preserving Möbius transformations are exactly the maps of the Riemann sphere covered here.  The orientation-reversing ones are obtained from these by complex conjugation.<ref>{{citation | first=Marcel| last=Berger| author-link=Marcel Berger| title=Geometry II| page=18.10|year=1987|publisher=Springer (Universitext)}}</ref>

The domain of Möbius transformations, i.e. {{tmath|1= \overline{\R^n} }}, is homeomorphic to the ''n''-dimensional sphere <math>S^n</math>. The canonical isomorphism between these two spaces is the [[Cayley transform]], which is itself a Möbius transformation of {{tmath|1= \overline{\R^{n + 1} } }}. This identification means that Möbius transformations can also be thought of as conformal isomorphisms of <math>S^n</math>. The ''n''-sphere, together with action of the Möbius group, is a geometric structure (in the sense of Klein's [[Erlangen program]]) called [[Conformal geometry#Möbius geometry|Möbius geometry]].<ref>{{citation|first1=Maks|last1=Akivis|first2=Vladislav|last2=Goldberg|title=Conformal differential geometry and its generalizations| publisher=Wiley-Interscience|year=1992}}</ref>