# Liouville's theorem (conformal mappings)

Liouville's theorem (conformal mappings)

In mathematics, Liouville's theorem is a theorem about conformal mappings in Euclidean space. It states that any conformal mapping on a domain of R"n", where "n" > 2, can be expressed as a composition of translations, similarities, orthogonal transformations and inversions: they are all Möbius transformations. This severely limits the variety of possible conformal mappings in R3 and higher-dimensional spaces.

By contrast, conformal mappings in R2 can be much more complicated - for example, all simply connected planar domains are conformally equivalent, by the Riemann mapping theorem.

Bibliography

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*springer|id=L/l059680|title=Liouville theorems|first=E.D.|last=Solomentsev|year=2001

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