Pseudo-Riemannian Symmetries on Heisenberg groups

The notion of $Gamma$-symmetric space is a natural generalization of the classical notion of symmetric space based on $Z_2$-grading of Lie algebras. In our case, we consider homogeneous spaces $G/H$ such that the Lie algebra $g$ of $G$ admits a $Gamma$-grading where $Gamma$ is a finite abelian group. In this work we study Riemannian metrics and Lorentzian metrics on the Heisenberg group $mathbb{H}_3$ adapted to the symmetries of a $Gamma$-symmetric structure on $mathbb{H}_3$. We prove that the classification of $z$-symmetric Riemannian and Lorentzian metrics on $mathbb{H}_3$ corresponds to the classification of left-invariant Riemannian and Lorentzian metrics, up to isometry. We study also the $Z_2^k$-symmetric structures on $G/H$ when $G$ is the $(2p+1)$-dimensional Heisenberg group for $k geq 1$. This gives examples of non riemannian symmetric spaces. When $k geq 1$, we show that there exists a family of flat and torsion free affine connections adapted to the $Z_2^k$-symmetric structures.