Editing Conformal map (section)

==In three or more dimensions==

===Riemannian geometry===
{{see also|Conformal geometry}}
In [[Riemannian geometry]], two [[Riemannian metric]]s <math>g</math> and <math>h</math> on a smooth manifold <math>M</math> are called '''conformally equivalent''' if <math> g = u h </math> for some positive function <math>u</math> on <math>M</math>. The function <math>u</math> is called the '''conformal factor'''.

A [[diffeomorphism]] between two Riemannian manifolds is called a '''conformal map''' if the pulled back metric is conformally equivalent to the original one.  For example, [[stereographic projection]] of a [[sphere]] onto the [[plane (mathematics)|plane]] augmented with a [[point at infinity]] is a conformal map.

One can also define a '''conformal structure''' on a smooth manifold, as a class of conformally equivalent [[Riemannian metric]]s.

===Euclidean space===
A [[Liouville's theorem (conformal mappings)|classical theorem]] of [[Joseph Liouville]] shows that there are far fewer conformal maps in higher dimensions than in two dimensions. Any conformal map from an open subset of [[Euclidean space]] into the same Euclidean space of dimension three or greater can be composed from three types of transformations: a [[homothetic transformation|homothety]], an [[isometry]], and a [[special conformal transformation]]. For [[Linear map|linear transformations]], a conformal map may only be composed of [[homothetic transformation|homothety]] and [[isometry]], and is called a [[conformal linear transformation]].