Conformal Map - Higher-dimensional Euclidean Space

Higher-dimensional Euclidean Space

A classical theorem of Joseph Liouville called Liouville's theorem shows the higher-dimensions have less varied conformal maps:

Any conformal map on a portion of Euclidean space of dimension greater than 2 can be composed from three types of transformation: a homothetic transformation, an isometry, and a special conformal transformation. (A special conformal transformation is the composition of a reflection and an inversion in a sphere.) Thus, the group of conformal transformations in spaces of dimension greater than 2 are much more restricted than the planar case, where the Riemann mapping theorem provides a large group of conformal transformations.