On Cohomogeneity One Linear Actions On Pseudo-euclidean Space $\mathbb{R}^{p,q}$

TL;DR

This paper classifies cohomogeneity one isometric linear actions on pseudo-Euclidean space, showing certain actions are not of this type and identifying conditions under which different actions are orbit-equivalent.

Contribution

It provides a detailed classification of cohomogeneity one actions on $ ext{R}^{p,q}$, including new results on orbit-equivalence and the role of parabolic subgroups.

Findings

Nilpotent factors of Iwasawa decompositions do not produce cohomogeneity one actions.

Certain subgroup actions have orbits that are orbit-equivalent outside a degenerate subspace.

Existence of cohomogeneity one actions with specific orbit structures on $ ext{R}^{p,q}$.

Abstract

The aim of this paper is to study cohomogeneity one isometric linear actions on the $p+q$-dimensional pseudo-Euclidean space $\mathbb{R}^{p,q}$. It is proved that the natural isometric action of the nilpotent factor of an Iwasawa decomposition of $SO(p,q)$ is not of cohomogeneity one. The orbits of cohomogeneity one actions of some subgroups of a maximal parabolic subgroup of the isometry group of $\mathbb{R}^{p,q}$ are determined and it is proved that there exist cohomogeneity one isometric actions on $\mathbb{R}^{p,q}$ which are orbit-equivalent on the complement of a $p$-dimensional degenerate subspace $\mathbb{W}^p$ of $\mathbb{R}^{p,q}$ and not orbit-equivalent on $\mathbb{W}p$.