  In this paper we derive necessary and sufficient conditions for a smooth
surface in Rn+1 to admit a local 1-quasiconformal parameterization by a domain
in Rn (n >= 3). We then apply these conditions to specific hypersurfaces such
as cylinders, paraboloids, and ellipsoids. As a consequence, we show that the
classical Liouville theorem about the rigidity of 1-quasiconformal maps between
domains in Rn with n >= 3 does not extend to embeddings of domains into a
higher dimensional space.
