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On completeness and dynamics of compact Brinkmann spacetimes

Lilia Mehidi, Abdelghani Zeghib

TL;DR

This paper analyzes the completeness and dynamical properties of Brinkmann spacetimes, focusing on compact and compactly Brinkmann-homogeneous manifolds. By developing a Cartan-type framework and constructing a core N invariant under a finite-index subgroup of the isometry group, the authors reduce global questions to almost locally homogeneous settings and study the flow of the parallel lightlike vector field V. They prove geodesic completeness in the compact and compactly Brinkmann-homogeneous cases and establish equicontinuity of the V-flow in key degeneracy and homogeneous scenarios, with a detailed treatment of flat bands, Cahen–Wallach spaces, and the degenerate closure foliation. The work combines synthetic completeness arguments with Cartan geometry, transversally Riemannian foliations, and lattice/holonomy analysis to derive a robust structural and dynamical picture of Brinkmann spacetimes and their cores. The results contribute to understanding Lorentzian completeness, rigidity phenomena, and the interaction between global topology and isometry groups in the Brinkmann setting.

Abstract

Brinkmann Lorentz manifolds are those admitting an isotropic parallel vector field. We prove geodesic completeness of the compact and also compactly homogeneous Brinkmann spaces. We also prove, partially, that their parallel vector field generates an equicontinuous flow.

On completeness and dynamics of compact Brinkmann spacetimes

TL;DR

This paper analyzes the completeness and dynamical properties of Brinkmann spacetimes, focusing on compact and compactly Brinkmann-homogeneous manifolds. By developing a Cartan-type framework and constructing a core N invariant under a finite-index subgroup of the isometry group, the authors reduce global questions to almost locally homogeneous settings and study the flow of the parallel lightlike vector field V. They prove geodesic completeness in the compact and compactly Brinkmann-homogeneous cases and establish equicontinuity of the V-flow in key degeneracy and homogeneous scenarios, with a detailed treatment of flat bands, Cahen–Wallach spaces, and the degenerate closure foliation. The work combines synthetic completeness arguments with Cartan geometry, transversally Riemannian foliations, and lattice/holonomy analysis to derive a robust structural and dynamical picture of Brinkmann spacetimes and their cores. The results contribute to understanding Lorentzian completeness, rigidity phenomena, and the interaction between global topology and isometry groups in the Brinkmann setting.

Abstract

Brinkmann Lorentz manifolds are those admitting an isotropic parallel vector field. We prove geodesic completeness of the compact and also compactly homogeneous Brinkmann spaces. We also prove, partially, that their parallel vector field generates an equicontinuous flow.
Paper Structure (56 sections, 18 theorems, 45 equations)

This paper contains 56 sections, 18 theorems, 45 equations.

Table of Contents

  1. Introduction
  2. Completeness
  3. Completeness results
  4. Case of Brinkmann spacetimes
  5. Reduction to an (almost) locally homogeneous situation
  6. Dynamics
  7. Dynamics in the locally homogeneous case
  8. Cahen-Wallach spaces
  9. Flat case
  10. A non-periodic example
  11. Organization of the article
  12. Brinkmann Geometry
  13. Preliminaries and local coordinates
  14. Transverse Riemannian structure
  15. Local coordinates
  16. ...and 41 more sections

Key Result

Theorem 1.4

A compactly Brinkmann-homogeneous spacetime is complete. In particular, both compact Brinkmann spacetimes and Brinkmann-homogeneous spacetimes are complete.

Theorems & Definitions (61)

  • Example 1.1
  • Definition 1.2
  • Definition 1.3
  • Theorem 1.4: Theorem \ref{['theorem-main: compact Brinkmann is complete']}
  • Theorem 1.5
  • Theorem 1.6
  • proof
  • proof
  • proof
  • Proposition 2.4
  • ...and 51 more