Lorentzian compact manifolds: isometries and geodesics

TL;DR

This paper studies four-dimensional compact Lorentzian manifolds, revealing properties of their geodesics, isometry groups, and actions of the Heisenberg group, contributing to the understanding of their geometric and symmetry structures.

Contribution

It characterizes geodesic periodicity and computes isometry groups for these manifolds, also demonstrating non-trivial isometric actions of the Heisenberg group.

Findings

All lightlike geodesics are periodic.

Existence of closed and non-closed spacelike and timelike geodesics.

Non-trivial isometric actions of is_3(\u211d) on certain nilmanifolds.

Abstract

In this work we investigate families of compact Lorentzian manifolds in dimension four. We show that every lightlike geodesic on such spaces is periodic, while there are closed and non-closed spacelike and timelike geodesics. Their isometry groups are computed. We also show that there is a non trivial action by isometries of $\Heis_3(\RR)$ on the nilmanifold $S^1\times (\Gamma_k \bsh \Heis_3(\RR))$ for $\Gamma_k$ a lattice of $\Heis_3(\RR)$.