On absolutely conformal mappings

Let Omega be a domain in R(n). It is proved that, if u is an element of C(1) (Omega; R(n)) and there holds the formula parallel to del u(x)parallel to(n) = n(n/2) vertical bar det del u(x)vertical bar in Omega, then for n >= 3 u is a restriction of a Mobius transformation, and for n = 2, u is an analytic function. This extends, partially, the well-known Lionville theorem ([6]), wich states that if u is an element of ACL(n)(Omega; R(n)), n >= 3, and the condition parallel to del u(x)parallel to(n) = n(n/2) det del u(x) is satisfied a.e. in Omega, then u is a restriction of a Mobius transformation.