In the mathematical field of analysis, quasiregular maps are a class of continuous maps between Euclidean spaces Rn of the same dimension or, more generally, between Riemannian manifolds of the same dimension, which share some of the basic properties with holomorphic functions of one complex variable.
This is a theorem of Joseph Liouville; relaxing the smoothness assumptions does not help, as proved by Yurii Reshetnyak.
[1] This suggests the search of a generalization of the property of conformality which would give a rich and interesting class of maps in higher dimension.
A differentiable map f of a region D in Rn to Rn is called K-quasiregular if the following inequality holds at all points in D: Here K ≥ 1 is a constant, Jf is the Jacobian determinant, Df is the derivative, that is the linear map defined by the Jacobi matrix, and ||·|| is the usual (Euclidean) norm of the matrix.
Many theorems about geometric properties of holomorphic functions of one complex variable have been extended to quasiregular maps.