Pseudo-Riemannian Symmetries on Heisenberg Groups

Abstract

Abstract The notion of Γ-symmetric space is a natural generalization of the classical notion of symmetric space based on Z2-grading on Lie algebras. 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 Riemannian and Lorentzian Zl-symmetric metrics on H3 corresponds to the classification of its left-invariant Riemannian and Lorentzian metrics, up to isometry. We study also the Z§-symmetric structures on G/H when G is the (2p + 1)-dimensional Heisenberg group. 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 Z§-symmetric structures.