Conformal mappings and CR mappings on the Engel group
type: Item
instance of: scholarly article
title: Conformal mappings and CR mappings on the Engel group
zbMATH Open document ID: 1204.32023
author: Wei Wang
author: Qingyan Wu
published in: Science in China. Series A
publication date: 2010-07-24T00:00:00Z
review text: We recall that a \textit{Carnot group} is a connected, simply connected nilpotent real Lie group, with graded Lie algebra \(\mathfrak g = V_1 \oplus V_2 \oplus \dots \oplus V_r\) with \([V_1, V_i] = V_{i+1}\) for any \(1 \leq i \leq r-1\). The \textit{Engel group} is the \(4\)-dimensional Carnot group \(G\) with Lie algebra \(\mathfrak g = V_1 \oplus V_2 \oplus V_3\) with \(\dim V_1 = 2\) and \(\dim V_2 = \dim V_3 = 1\). Once a complex structure \(J_o: V_1 \to V_1\) and a \(J_o\)-invariant Euclidean inner product \(g_o\) on \(V_1\) are fixed, the Engel group \(G\) is naturally endowed with: a) the \(G\)-invariant CR structure \((\mathcal D, J)\) where \(\mathcal D \subset TG\) is the invariant distribution with \(\mathcal D_e = V_1\) and \(J\) is the invariant family of complex structures on the spaces \(\mathcal D_p\), \(p \in G\), with \(J|_{\mathcal D_e} = J_o\); b) the invariant family \(g\) of Euclidean inner products on the spaces \(\mathcal D_p\), \(p \in G\), with \(g|_{\mathcal D_e \times \mathcal D_e} = g_o\); c) the invariant Carnot-Charathéodory pseudo-distance \(d_c(p, q)\) between points \(p,q \in G\), given by the infimum over the lengths w.r.t. \(g\) of the absolutely continuous curves that join \(p\) to \(q\) and with all tangent vectors in \(\mathcal D\). The authors show that any diffeomorphism \(f: G \longrightarrow G\) which is conformal w.r.t. the pseudo-distance \(d_c\) is either CR or anti-CR automorphism of \((G, \mathcal D, J)\) (i.e. so that \(f_*(\mathcal D) = \mathcal D\) and \(f_* J = \pm J\)). Since \(\text{Aut}^0(G, \mathcal D, J)\) is known, this result explicitly determines the identity component of the conformal group of \((G, d_c)\).
Mathematics Subject Classification ID: 32V05
Mathematics Subject Classification ID: 35R03
Mathematics Subject Classification ID: 22E25
Mathematics Subject Classification ID: 53A30
zbMATH DE Number: 5760403
zbMATH Keywords: Engel group
zbMATH Keywords: CR automorphism
zbMATH Keywords: quasi-conformal mapping
reviewed by: Andrea Spiro
MaRDI profile type: Publication
full work available at URL: s11425-009-0177-5
OpenAlex ID: W2041465498
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DOI: 10.1007/S11425-009-0177-5
description: scientific article; zbMATH DE number 5760403