Cohomogeneity one actions on Minkowski spaces

Abstract

We study isometric cohomogeneity one actions on the $(n+1)$-dimensional Minkowski space $\mathbb {L}^{n+1}$ up to orbit-equivalence. We give examples of isometric cohomogeneity one actions on $\mathbb {L}^{n+1}$ whose orbit spaces are non-Hausdorff. We show that there exist isometric cohomogeneity one actions on $\mathbb {L}^{n+1}$, $n \ge 3$, which are orbit-equivalent on the complement of an n-dimensional degenerate subspace $\mathbb {W}^n$ of $\mathbb {L}^{n+1}$ and not orbit-equivalent on $\mathbb {W}^n$. We classify isometric cohomogeneity one actions on $\mathbb {L}^2$ and $\mathbb {L}^3$ up to orbit-equivalence.