3-folds CR-embedded in 5-dimensional real hyperquadrics

2018/08/31 by Curtis Porter · 1 citation
Mathematics · Physics and Astronomy · #Advanced Differential Geometry Research #Combinatorics #Congruence (geometry) #Geodesic #Geodesics in general relativity #Geometric Analysis and Curvature Flows #Geometry #Holomorphic and Operator Theory #Homogeneous #Homogeneous space #Lie group #Mathematical physics #Mathematics #Minkowski space #Physics #Pure mathematics #Quantum mechanics #Spacetime #math.DG

paper · pdf · doi:10.1016/j.geomphys.2021.104107

published in Journal of Geometry and Physics 163, 104107 (Elsevier BV) · Substantial revision of Levi-nondegenerate classification, both intrinsic and extrinsic, for clarity; Levi-flat case now included; 32 pages

arxiv created 2020/05/29 · openalex publication_date 2021/01/13 · arxiv updated 2021/02/23 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/05

Abstract

E. Cartan's method of moving frames is applied to 3-dimensional manifolds M which are CR-embedded in 5-dimensional real hyperquadrics Q in order to classify M up to CR symmetries of Q given by the action of one of the Lie groups SU(3,1) or SU(2,2). In the latter case, the CR structure of M derives from a shear-free null geodesic congruence on Minkowski spacetime, and the relationship to relativity is discussed. In both cases, we compute which homogeneous CR 3-folds appear in Q.

Citations

• On Cartan’s method of Lie groups and moving frames as applied to uniqueness and existence questions in differential geometry
• Spinors and Space-Time
• Real hypersurfaces in complex manifolds
• Gravitational waves in general relativity. VI. The outgoing radiation condition
• Gravitational Field of a Spinning Mass as an Example of Algebraically Special Metrics