2015/03/04 by Gestur Ólafsson, Ólafsson, Gestur, Raúl Quiroga-Barranco +1
Mathematics · #53C24 #53C50 #57S20 #Advanced Algebra and Geometry #Advanced Operator Algebra Research #Differential Geometry (math.DG) #FOS: Mathematics #Geometry and complex manifolds #Group Theory (math.GR)
paper · pdf · doi:10.48550/arxiv.1503.01483
openalex publication_date 2015/03/04 · openalex created_date 2022/10/02 · openalex updated_date 2026/08/01
Denote by widetilde\U(p,q) the universal covering group of\n\U(p,q), the linear group of isometries of the pseudo-Hermitian space\n\ℂp,q of signature p,q. Let M be a connected analytic complete\npseudo-Riemannian manifold that admits an isometric\n widetilde\U(p,q)-action and that satisfies \dim M \≤ n(n+2)\nwhere n = p+q. We prove that if the action of widetilde\SU(p,q)\n(the connected derived group of widetilde\U(p,q)) has a dense\norbit and the center of widetilde\U(p,q) acts non-trivially, then\nM is an isometric quotient of manifolds involving simple Lie groups with\nbi-invariant metrics. Furthermore, the widetilde\U(p,q)-action is\nlifted to widetildeM to natural actions on the groups involved. As a\nparticular case, we prove that when widetildeM is not a pseudo-Riemannian\nproduct, then its geometry and widetilde\U(p,q)-action are\nobtained from one of the symmetric pairs ( mathfraksu(p,q+1),\n mathfraku(p,q)) or ( mathfraksu(p+1,q), mathfraku(p,q)).\n