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Completeness of compact Locally symmetric Lorentz manifolds

Souheib Allout, Malek Hanounah

TL;DR

This work resolves geodesic completeness for compact locally symmetric Lorentz manifolds by exploiting a product decomposition of symmetric spaces into a Lorentz factor $X^L$ and a Riemannian factor $Y$, together with natural foliations and a careful analysis of the developing map. The authors rule out obstructions coming from de Sitter indecomposables via the Calabi–Markus phenomenon, address the AdS/dS and flat cases with product-geometry convexity arguments, and prove a no-cocompact-lattice result for half-Minkowski models, ultimately establishing completeness in all cases. The approach blends symmetric-space structure, foliation theory, and discrete-group dynamics (Auslander, syndetic hulls, unimodularity) to cover the full range of indecomposable Lorentz factors, including Cahen–Wallach, timelike, and flat scenarios. The findings clarify the role of transversely modeled factors and provide a robust framework for future investigations into completeness questions in higher-signature and foliated semi-Riemannian geometries. The results have implications for Markus-type conjectures and the broader understanding of global properties of locally symmetric Lorentzian spaces.

Abstract

We show that compact locally symmetric Lorentz manifolds are geodesically complete.

Completeness of compact Locally symmetric Lorentz manifolds

TL;DR

This work resolves geodesic completeness for compact locally symmetric Lorentz manifolds by exploiting a product decomposition of symmetric spaces into a Lorentz factor and a Riemannian factor , together with natural foliations and a careful analysis of the developing map. The authors rule out obstructions coming from de Sitter indecomposables via the Calabi–Markus phenomenon, address the AdS/dS and flat cases with product-geometry convexity arguments, and prove a no-cocompact-lattice result for half-Minkowski models, ultimately establishing completeness in all cases. The approach blends symmetric-space structure, foliation theory, and discrete-group dynamics (Auslander, syndetic hulls, unimodularity) to cover the full range of indecomposable Lorentz factors, including Cahen–Wallach, timelike, and flat scenarios. The findings clarify the role of transversely modeled factors and provide a robust framework for future investigations into completeness questions in higher-signature and foliated semi-Riemannian geometries. The results have implications for Markus-type conjectures and the broader understanding of global properties of locally symmetric Lorentzian spaces.

Abstract

We show that compact locally symmetric Lorentz manifolds are geodesically complete.

Paper Structure

This paper contains 25 sections, 23 theorems, 19 equations.

Table of Contents

  1. Introduction
  2. Calabi--Markus phenomenon
  3. Beyond the Lorentz signature
  4. Symmetric spaces
  5. Splitting of the isometry group
  6. Injectivity of the developing map
  7. Natural foliations
  8. A natural action on $\widetilde{M}$
  9. Product structure of the universal cover
  10. The injectivity
  11. The case of ${\operatorname{AdS}}$ or ${\operatorname{dS}}$ Lorentz factor
  12. The de Sitter case: Calabi--Markus phenomenon
  13. The foliated setting
  14. The case of flat Lorentz factor
  15. Algebraic formulation
  16. ...and 10 more sections

Key Result

Theorem 1.1

A compact locally symmetric Lorentz manifold is geodesically complete.

Theorems & Definitions (43)

  • Theorem 1.1
  • Proposition 1.2
  • Definition 2.1
  • Remark 2.4: Cahen--Wallach spaces
  • Proposition 2.5
  • proof
  • Lemma 3.1
  • proof
  • Corollary 3.2
  • Corollary 3.3
  • ...and 33 more