Pseudo-Riemannian Symmetries on Heisenberg groups

2014/01/27 by Michel Goze, Goze, Michel, Paola Piu +3
Mathematics · Physics and Astronomy · #17B70 #22F30 #53C30 #53C35 #Advanced Differential Geometry Research #Differential Geometry (math.DG) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Geometry and complex manifolds

paper · pdf · doi:10.48550/arxiv.1401.6802

openalex publication_date 2014/01/27 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

The notion of Γ-symmetric space is a natural generalization of the classical notion of symmetric space based on \Z₂-grading of Lie algebras. In our case, we consider homogeneous spaces G/H such that the Lie algebra \g of G admits a Γ-grading where Γ is a finite abelian group. In this work we study Riemannian metrics and Lorentzian metrics on the Heisenberg group ℍ₃ adapted to the symmetries of a Γ-symmetric structure on ℍ₃. We prove that the classification of \z-symmetric Riemannian and Lorentzian metrics on ℍ₃ corresponds to the classification of left-invariant Riemannian and Lorentzian metrics, up to isometry. We study also the \Z₂^k-symmetric structures on G/H when G is the (2p+1)-dimensional Heisenberg group for k ≥ 1. This gives examples of non riemannian symmetric spaces. When k ≥ 1, we show that there exists a family of flat and torsion free affine connections adapted to the \Z₂^k-symmetric structures.

Related

• Pseudo-Riemannian Symmetries on Heisenberg group ℍ₃
• On Riemannian non symmetric spaces and flag manifolds
• On symmetries of the Heisenberg group
• A classification of left-invariant Lorentzian metrics on some nilpotent Lie groups
• A classification of left-invariant pseudo-Riemannian metrics on some nilpotent Lie groups