Pseudo-Riemannian Symmetries on Heisenberg group $\mathbb{H}_{3}$

TL;DR

This paper classifies Riemannian and Lorentzian metrics on the Heisenberg group that are compatible with $ ext{Z}_2^2$-symmetries, providing examples of non-symmetric Lorentzian homogeneous spaces.

Contribution

It introduces a classification of $ ext{Z}_2^2$-symmetric metrics on $ ext{H}_3$, linking them to left-invariant metrics and expanding the understanding of non-symmetric Lorentzian homogeneous spaces.

Findings

Classification of $ ext{Z}_2^2$-symmetric metrics on $ ext{H}_3$

Correspondence with left-invariant metrics

Examples of non-symmetric Lorentzian homogeneous spaces

Abstract

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 $\z_2^2$-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.