Pseudo-Riemannian Symmetries on Heisenberg group ${\mathbb{H}}_3$

The notion of $\mathrm{\Gamma }$-symmetric space is a natural generalization of the classical notion of symmetric space based on ${\text{z}}_2$-grading of Lie algebras. In our case, we consider homogeneous spaces $G\mathrm{/}H$ such that the Lie algebra $\text{g}$ of $G$ admits a $\mathrm{\Gamma }$-grading where $\mathrm{\Gamma }$ 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 $\mathrm{\Gamma }$-symmetric structure on ${\mathbb{H}}_3$. We prove that the classification of ${\text{z}}_2^2$-symmetric Riemannian and Lorentzian metrics on ${\mathbb{H}}_3$ corresponds to the classification of left invariant Riemannian and Lorentzian metrics, up to isometries. This gives examples of non-symmetric Lorentzian homogeneous spaces.