Isometry groups and geodesic foliations of Lorentz manifolds. Part I: Foundations of Lorentz dynamics

TL;DR

This paper explores the dynamics of Lorentz transformations on compact Lorentz manifolds, focusing on isometry groups and geodesic foliations, laying foundational concepts for classifying such manifolds.

Contribution

It introduces the study of affine and Lorentz transformations' dynamics and their connection to geodesic foliations, providing foundational insights for future classification work.

Findings

Relationship between Lorentz transformations and geodesic foliations established

Framework for analyzing isometry group actions on Lorentz manifolds developed

Preliminary steps towards classifying Lorentz manifolds with non-compact isometry groups

Abstract

This is the first part of a series on non-compact groups acting isometrically on compact Lorentz manifolds. This subject was recently investigated by many authors. In the present part we investigate the dynamics of affine, and especially Lorentz transformations. In particular we show how this is related to geodesic foliations. The existence of geodesic foliations was (very succinctly) mentioned for the first time by D'Ambra and Gromov, who suggested that this may help in the classification of compact Lorentz manifolds with non-compact isometry groups. In the Part II of the series, a partial classification of compact Lorentz manifolds with non-compact isometry group will be achieved with the aid of geometrical tools along with the dynamical ones presented here.