Contact and conformal maps on Iwasawa N groups

Cowling, Michael ; De Mari, Filippo ; Korányi, Adam ; Reimann, Hans Martin

Atti della Accademia nazionale dei Lincei. Rendiconti Lincei. Matematica e applicazioni, Série 9, Tome 13 (2002) no. 3-4, pp. 219-232

Résumé

The action of the conformal group $O(1,n + 1)$ on $\mathbb{R}^{n} \cup \{\infty\}$ may be characterized in differential geometric terms, even locally: a theorem of Liouville states that a $C^{4}$ map between domains $U$ and $V$ in $\mathbb{R}^{n}$ whose differential is a (variable) multiple of a (variable) isometry at each point of $U$ is the restriction to $U$ of a transformation $x \rightarrow g \cdot x$, for some $g$ in $O(1,n + 1)$. In this paper, we consider the problem of characterizing the action of a more general semisimple Lie group $G$ on the space $G/P$ , where $P$ is a parabolic subgroup. We solve this problem for the cases where $G$ is $SL(3,\mathbb{R})$ or $Sp(2,\mathbb{R})$ and $P$ is a minimal parabolic subgroup.