Conformal mappings and CR mappings on the Engel group
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 \((G, d_c)\).
scientific article; zbMATH DE number 5760403