Cohomogeneity One Three Dimensional Anti de Sitter Space, Proper and Nonproper Actions

TL;DR

This paper classifies Lie group actions of cohomogeneity one on 3D anti de Sitter space, analyzing orbit structures and causal properties for both proper and nonproper actions.

Contribution

It provides a comprehensive classification of Lie group actions on ${f adS_3}$ and characterizes the causal nature of their orbits, including proper and nonproper cases.

Findings

Classified all closed, connected Lie groups acting on ${f adS_3}$ by cohomogeneity one.

Determined the causal characters of orbits and orbit spaces for proper and nonproper actions.

Showed that proper actions have no exceptional orbits and principal orbits share the same causal character.

Abstract

In this paper we give a classification of closed and connected Lie groups, up to conjugacy in $Iso({\bf adS_3})$, acting by cohomogeneity one on the three dimensional anti de sitter space ${\bf adS_3}$. Then we determine causal characters of the orbits and the orbit spaces, up to homeomorphism, in both cases, proper and nonproper actions. When the action is proper, we show that there is no exceptional orbit and causal characters of the principal orbits are the same.