On low-dimensional manifolds with isometric SO0(p, q)-actions

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) ≥ dim(G) + m(\( \mathfrak{g} \)) when M admits an isometric G-action with a dense orbit. The Main Theorem considers the case \( G = {\widetilde {\text{SO}}_0}\left( {p,q} \right) \), providing an explicit description of M when the bound is achieved. In such a case, M is (up to a finite covering) the quotient by a lattice of either \( {\widetilde {\text{SO}}_0}\left( {p + 1,q} \right) \) or \( {\widetilde {\text{SO}}_0}\left( {p,q + 1} \right) \).