Pseudo-Riemannian Symmetries on Heisenberg groups

الأصل · EN

The notion of Γ-symmetric space is a natural generalization of the classical notion of symmetric space based on ₂-grading of Lie algebras. In our case, we consider homogeneous spaces G/H such that the Lie algebra 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 -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 ₂ᵏ-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 ₂ᵏ-symmetric structures.