te ariā a Liouville

In mathematics, Liouville's theorem, proved by Joseph Liouville in 1850, is a rigidity theorem about conformal mappings in Euclidean space. It states that every smooth conformal mapping on a domain of Rⁿ, where n > 2, can be expressed as a composition of translations, similarities, orthogonal transformations and inversions: they are Möbius transformations. This theorem 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.