[2][3] This theorem severely limits the variety of possible conformal mappings in R3 and higher-dimensional spaces.
Generalizations of the theorem hold for transformations that are only weakly differentiable (Iwaniec & Martin 2001, Chapter 5).
A weak solution of this system is defined to be an element f of the Sobolev space W1,nloc(Ω, Rn) with non-negative Jacobian determinant almost everywhere, such that the Cauchy–Riemann system holds at almost every point of Ω. Liouville's theorem is then that every weak solution (in this sense) is a Möbius transformation, meaning that it has the form where a, b are vectors in Rn, α is a scalar, A is a rotation matrix, ε = 0 or 2, and the matrix in parentheses is I or a Householder matrix (so, orthogonal).
Equivalently stated, any quasiconformal map of a domain in Euclidean space that is also conformal is a Möbius transformation.
Equality of the two dimensions holds exactly when the conformal manifold is isometric with the n-sphere or projective space.