Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Completeness of closed Kleinian flat Pseudo-Riemannian Manifolds of Signature (2,2)

Farid Diaf, Blandine Galiay, Malek Hanounah

Abstract

Let $\mathbb{R}^{2,2}$ denote the model space of flat pseudo-Riemannian manifolds of signature $(2,2)$. We prove that the only domain divisible by a discrete subgroup of the isometry group of $\mathbb{R}^{2,2}$ is $\mathbb{R}^{2,2}$ itself. In the Kleinian setting, this provides the first completeness theorem of closed flat pseudo-Riemannian manifolds beyond the Euclidean and Lorentzian cases. Along the proof, we show two results of independent interest. The first is a geometric reduction for certain divisible domains of affine space. The second concerns the existence of syndetic hulls in semidirect products $R \ltimes G$, where $G$ is a homothety Lie group. This construction generalizes earlier constructions in affine geometry due to Carrière and Dal'bo.

Completeness of closed Kleinian flat Pseudo-Riemannian Manifolds of Signature (2,2)

Abstract

Let denote the model space of flat pseudo-Riemannian manifolds of signature . We prove that the only domain divisible by a discrete subgroup of the isometry group of is itself. In the Kleinian setting, this provides the first completeness theorem of closed flat pseudo-Riemannian manifolds beyond the Euclidean and Lorentzian cases. Along the proof, we show two results of independent interest. The first is a geometric reduction for certain divisible domains of affine space. The second concerns the existence of syndetic hulls in semidirect products , where is a homothety Lie group. This construction generalizes earlier constructions in affine geometry due to Carrière and Dal'bo.
Paper Structure (30 sections, 67 theorems, 83 equations, 1 table)

This paper contains 30 sections, 67 theorems, 83 equations, 1 table.

Table of Contents

  1. Introduction
  2. Main result
  3. Strategy of the proof
  4. Reduction result
  5. Quotient geometry
  6. Organization of the paper
  7. Acknowledgments
  8. Preliminaries on affine geometry
  9. Affine manifolds
  10. Algebraic and cohomological constraints
  11. Completeness results
  12. Syndetic hulls
  13. Reduction of the flat affine geometry
  14. General reduction result
  15. General reduction when the linear part is contained in a semisimple Lie subgroup
  16. ...and 15 more sections

Key Result

Theorem 1

Let $\Omega$ be a domain of $\mathbb{R}^{2,2}$ and let $\Gamma\le {\operatorname{SO}}(2,2)\ltimes \mathbb{R}^{2,2}$ be a discrete group acting freely, properly discontinuously, and cocompactly on $\Omega$. Then $\Omega=\mathbb{R}^{2,2}$.

Theorems & Definitions (136)

  • Theorem 1
  • Proposition 2
  • Theorem 3
  • Definition 2.1
  • Theorem 2.2
  • Theorem 2.3: GH_cohomology
  • Theorem 2.4: Nilpotent_complete
  • Theorem 2.5
  • Definition 2.6
  • Theorem 2.7
  • ...and 126 more