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 (in n dimensions).[1] [2] 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.
Generalizations of the theorem hold for transformations that are only weakly differentiable . The focus of such a study is the non-linear Cauchy–Riemann system that is a necessary and sufficient condition for a smooth mapping to be conformal:
DfTDf=\left|\detDf\right|2/nI
f(x)=b+
\alphaA(x-a) | |
|x-a|\varepsilon |
, Df=
\alphaA | \left(I-\varepsilon | |
|x-a|\varepsilon |
x-a | |
|x-a| |
(x-a)T | |
|x-a| |
\right),
Similar rigidity results (in the smooth case) hold on any conformal manifold. The group of conformal isometries of an n-dimensional conformal Riemannian manifold always has dimension that cannot exceed that of the full conformal group . Equality of the two dimensions holds exactly when the conformal manifold is isometric with the n-sphere or projective space. Local versions of the result also hold: The Lie algebra of conformal Killing fields in an open set has dimension less than or equal to that of the conformal group, with equality holding if and only if the open set is locally conformally flat.