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 $F:{\mathbb{R}}^{n+1}\to {\mathbb{R}}^n$ is conformal and bounded, then it is a constant mapping, provided that $n\ge 2$}~? We present arguments confirming the validity of such a Liouville-type theorem.

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