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On closed geodesics in Lorentz manifolds

Souheib Allout, Abderrahmane Belkacem, Abdelghani Zeghib

Abstract

We construct compact Lorentz manifolds without closed geodesics.

On closed geodesics in Lorentz manifolds

Abstract

We construct compact Lorentz manifolds without closed geodesics.
Paper Structure (11 sections, 13 theorems, 14 equations, 1 figure)

This paper contains 11 sections, 13 theorems, 14 equations, 1 figure.

Table of Contents

  1. Introduction
  2. The homogeneous case: proof of Theorem \ref{['homogeneous']}
  3. The geodesic equation for left invariant metrics
  4. Dynamics of the geodesic flow
  5. Compact quotients
  6. No closed geodesics: proof of Theorem \ref{['non closed']}
  7. ${\sf{SL}}(2,\mathbb{R})$ case.
  8. ${\mathsf{Sol}}$ case
  9. The ${\mathsf{Sol}} \times {\sf{Euc}}$ case: proof of Theorem \ref{['non periodic']}
  10. The product ${\mathsf{Sol}} \times {\sf{Euc}}$
  11. A compact quotient without closed or weakly closed geodesics

Key Result

Theorem 1.1

There exists a left invariant metric on $G$, of signature $(2,4)$, and a cocompact lattice $\Gamma\subset G$ such that $\Gamma\backslash G$ is without closed or weakly closed geodesics.

Figures (1)

  • Figure 1: The dynamics on $\mathbb{P}^+(\sl(2,\mathbb{R}))$

Theorems & Definitions (24)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Lemma 2.1
  • proof
  • Corollary 2.2
  • Remark 3.1
  • Lemma 4.1
  • proof
  • Corollary 4.2
  • ...and 14 more