Conformality in the sense of Gromov and a generalized Liouville theorem

Abstract

M.Gromov extended the concepts of conformal and quasiconformal mapping to the mappings acting between the manifolds of different dimensions. For instance, any entire holomorphic function $ f: \Cn \to {\mathbb C}$ defines a mapping conformal in the sense of Gromov. In this connection Gromov addressed a natural question: which facts of the classical theory apply to these mappings? In particular is it true that {\em If the mapping $ F: \R^{n + 1} \to \R^{n}$ is conformal and bounded, then it is a constant mapping, provided that $ n \geq 2 $}~? We present arguments confirming the validity of such a Liouville-type theorem.