Closed geodesics on compact Lorentzian solvmanifolds

TL;DR

This paper investigates the conditions under which geodesics are closed on compact Lorentzian solvmanifolds, revealing that lattice choices significantly influence geodesic closedness and exploring isometry groups in higher dimensions.

Contribution

It provides new criteria for closed geodesics on Lorentzian solvmanifolds and analyzes how lattice structures affect geodesic properties and symmetries.

Findings

Conditions for closed lightlike geodesics depend on the lattice.

Not all lightlike geodesics are closed in some 4D examples.

Isometry groups are computed for certain six-dimensional cases.

Abstract

The aim of this work is the study of geodesics on Lorentzian homogeneous spaces of the form $M=G/\Lambda$, where $G$ is a solvable Lie group endowed with a bi-invariant Lorentzian metric and $\Lambda < G$ is a cocompact lattice. Conditions to assert closedness of light, time or spacelike geodesics on the compact quotient spaces are given. This study implicitly requires additional information about the lattices in each case. We found conditions for which every lightlight geodesic on the quotient space is closed. And more important, this situation depends on the lattice. Moreover, even in dimension four, there are examples of compact solvmanifolds for which not every lightlike geodesic is closed. For time and spacelike geodesics, the conclusion are different. Finally, we study isometry groups of those compact spaces and show some computations in dimension six.