Editing Conformal geometry (section)

====Euclidean space====
[[Image:Conformal grid before Möbius transformation.svg|thumb|right|A coordinate grid prior to a Möbius transformation]]
[[Image:Conformal grid after Möbius transformation.svg|thumb|right|The same grid after a Möbius transformation]]
The group of conformal symmetries of the quadratic form

:<math>q(z,\bar{z}) = z\bar{z} </math>

is the group {{nowrap|1=GL<sub>1</sub>('''C''') = '''C'''<sup>×</sup>}}, the [[multiplicative group]] of the complex numbers.  Its Lie algebra is {{nowrap|1='''gl'''<sub>1</sub>('''C''') = '''C'''}}.

Consider the (Euclidean) [[complex plane]] equipped with the metric

:<math>g = dz \, d\bar{z}.</math>

The infinitesimal conformal symmetries satisfy
#<math>\mathbf{L}_X \, dz = f(z) \, dz</math>
#<math>\mathbf{L}_X \, d\bar{z} = f(\bar{z}) \, d\bar{z} ,</math>
where ''f'' satisfies the [[Cauchy–Riemann equation]], and so is [[Holomorphic function|holomorphic]] over its domain.  (See [[Witt algebra]].)

The conformal isometries of a domain therefore consist of holomorphic self-maps.  In particular, on the conformal compactification &ndash; the [[Riemann sphere]] &ndash; the conformal transformations are given by the [[Möbius transformation]]s

:<math>z \mapsto \frac{az+b}{cz+d}</math>

where {{nowrap|''ad'' &minus; ''bc''}} is nonzero.