Isometric actions of Lie subgroups of the Moebius group. (English) Zbl 1073.53016

Boubel, Charles; Zeghib, Abdelghani

Nonlinearity 17, No. 5, 1677-1688 (2004).

The following theorem is considered: Let \(G\) be a non-compact connected Lie subgroup of \(\text{Isom}(\mathbb H^n)\) which does not fix any point at infinity (i.e. on \(\partial \mathbb H^n \simeq\mathbb S^{n-1})\). Then, up to conjugacy, \(G\) preserves a hyperbolic subspace \(\mathbb H^d\) with \(1\leq d\leq n\) and contains \(O^0(d,1)\).
The authors give a geometric – essentially dynamic – proof and an application to the holonomy of Lorentzian manifolds.

Reviewer: Liu Hui-Li (Shenyang)

MSC:

53A30	Conformal differential geometry (MSC2010)
37C85	Dynamics induced by group actions other than \(\mathbb{Z}\) and \(\mathbb{R}\), and \(\mathbb{C}\)
57S20	Noncompact Lie groups of transformations
53C30	Differential geometry of homogeneous manifolds
53C29	Issues of holonomy in differential geometry

Keywords:

isometric action; hyperbolic subspace; Lie subgroup; Möbius group