On low-dimensional manifolds with isometric Ũ (p, q) - Actions
Denote by Ũ(p, q) the universal covering group of Ũ(p, q), the linear group of isometries of the pseudo-Hermitian space Cp,q of signature p, q. Let M be a connected analytic complete pseudo-Riemannian manifold that admits an isometric Ũ(p, q)-action and that satisfies dim M ≤ n(n + 2) where n = p + q. We prove that if the action of SU(p, q) (the connected derived group of eU(p, q)) has a dense orbit and the center of eU(p, q) acts non-trivially, then M is an isometric quotient of manifolds involving simple Lie groups with bi-invariant metrics. Furthermore, the Ũ(p, q)-action is lifted to M∼to natural actions on the groups involved. As a particular case, we prove that when M∼is not a pseudo-Riemannian product, then its geometry and Ũ(p, q)-action are obtained from one of the symmetric pairs (su(p, q + 1), u(p, q)) or (su(p + 1, q), u(p, q)).
Publication Source (Journal or Book title)
Asian Journal of Mathematics
Ólafsson, G., & Quiroga-Barranco, R. (2017). On low-dimensional manifolds with isometric Ũ (p, q) - Actions. Asian Journal of Mathematics, 21 (5), 873-908. https://doi.org/10.4310/AJM.2017.v21.n5.a5