On low-dimensional manifolds with isometric $\mathrm{SO}_0(p,q)$-actions

TL;DR

This paper characterizes low-dimensional pseudo-Riemannian manifolds with dense isometric actions by certain non-compact Lie groups, explicitly describing the manifolds when the minimal dimension bound is attained.

Contribution

It provides a classification of manifolds with isometric SO_0(p,q)-actions that achieve the minimal dimension bound, identifying them as quotients by lattices in larger orthogonal groups.

Findings

Dimension bound for manifolds with dense G-action

Explicit description of manifolds achieving the bound

Identification of manifolds as quotients by lattices in SO_0(p+1,q) or SO_0(p,q+1)

Abstract

Let $G$ be a non-compact simple Lie group with Lie algebra $\mathfrak{g}$. Denote with $m(\mathfrak{g})$ the dimension of the smallest non-trivial $\mathfrak{g}$-module with an invariant non-degenerate symmetric bilinear form. For an irreducible finite volume pseudo-Riemannian analytic manifold $M$ it is observed that $\dim(M) \geq \dim(G) + m(\mathfrak{g})$ when $M$ admits an isometric $G$-action with a dense orbit. The Main Theorem considers the case $G = \widetilde{\mathrm{SO}}_0(p,q)$ providing an explicit description of $M$ when the bound is achieved. In such case, $M$ is (up to a finite covering) the quotient by a lattice of either $\widetilde{\mathrm{SO}}_0(p+1,q)$ or $\widetilde{\mathrm{SO}}_0(p,q+1)$.