Contact and Conformal Maps in

Abstract

When $n \geqslant 3,$ the action of the conformal group O(1, n+1) on ${\mathbb R}^n \cup\{\infty\}$ may be characterized in simple differential geometric terms, even locally: a theorem of Liouville states that a C⁴ map between domains ${\cal U}$ and ${\cal V}$ in ${\mathbb R}^n$ whose differential is a (variable) multiple of a (variable) isometry at each point of ${\cal U}$ is the restriction to ${\cal U}$ of a transformation x g·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 minimal parabolic subgroup.