Asian Journal of Mathematics

Volume 21 (2017)

Number 5

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

Pages: 873 – 908

DOI: https://dx.doi.org/10.4310/AJM.2017.v21.n5.a5

Authors

Gestur Ólafsson (Department of Mathematics, Louisiana State University, Baton Rouge, La., U.S.A.)

Raul Quiroga-Barranco (Centro de Investigación en Matemáticas, Guanajuato, Mexico)

Abstract

Denote by $\widetilde{\mathrm{U}} (p, q)$ the universal covering group of $\mathrm{U} (p, q)$, the linear group of isometries of the pseudo-Hermitian space $\mathbb{C}^{p, q}$ of signature $p, q$. Let $M$ be a connected analytic complete pseudo-Riemannian manifold that admits an isometric $\widetilde{\mathrm{U}} (p, q)$-action and that satisfies $\mathrm{dim} \: M \leq n(n + 2)$ where $n = p + q$. We prove that if the action of $\widetilde{\mathrm{SU}} (p, q)$ (the connected derived group of $\widetilde{\mathrm{U}} (p, q)$) has a dense orbit and the center of $\widetilde{\mathrm{U}} (p, q)$ acts non-trivially, then $M$ is an isometric quotient of manifolds involving simple Lie groups with bi-invariant metrics. Furthermore, the $\widetilde{\mathrm{U}} (p, q)$-action is lifted to $\widetilde{M}$ to natural actions on the groups involved. As a particular case, we prove that when $\widetilde{M}$ is not a pseudo-Riemannian product, then its geometry and $\widetilde{\mathrm{U}} (p, q)$-action are obtained from one of the symmetric pairs $(\mathfrak{su}(p, q + 1), \mathfrak{u}(p, q))$ or $(\mathfrak{su}(p + 1, q), \mathfrak{u}(p, q))$.

Received 3 November 2015

Accepted 15 April 2016

Published 9 February 2018