Pseudo-Riemannian Symmetries on Heisenberg groups

Abstract

The notion of $\Gamma$-symmetric space is a natural generalization of the classical notion of symmetric space based on ${\mathbb{Z}}_2$-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 ${\mathbb{Z}}_2^k$-symmetric structures on $G/H$ when $G$ is the $(2p+1)$-dimensional Heisenberg group for $k\ge 1$. This gives examples of non riemannian symmetric spaces. When $k\ge 1$, we show that there exists a family of flat and torsion free affine connections adapted to the ${\mathbb{Z}}_2^k$-symmetric structures.

17 pages. arXiv admin note: text overlap with arXiv:1201.0447