On cohomogeneity one linear actions on pseudo-Euclidean space Rp,q

The aim of this paper is to study cohomogeneity one isometric linear actions on the p + q-dimensional pseudo-Euclidean space R-p,R-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 R-p,R-q are determined and it is proved that there exist cohomogeneity one isometric actions on R-p,R-q which are orbit-equivalent on the complement of a p-dimensional degenerate subspace W-p of R-p,R-q and not orbit-equivalent on W-p. (C) 2019 Elsevier B.V. All rights reserved.