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Introduction to Kundt spaces

Lilia Mehidi, Abdelghani Zeghib

TL;DR

Kundt spaces are characterized geometrically by a non-singular lightlike geodesic field whose orthogonal distribution is integrable and totally geodesic. The paper develops a coordinate-free framework, linking twist-free, shear-free, and divergence-free conditions to a transversally Riemannian foliation, and shows these equivalences yield a local metric form that encompasses Brinkmann spaces, pp-waves, plane waves, and Siklos spaces. It classifies examples, analyzes how Lie-group actions and Anosov dynamics generate Kundt structures, and studies global/topological restrictions in low dimensions, including surfaces and 3-manifolds, as well as high-regularity vs low-regularity scenarios. The work also situates Kundt spaces within a hierarchy of Lorentzian geometries and demonstrates that dynamical properties of the isometry group can induce locally Kundt or globally Kundt structures, with implications for both mathematical physics and differential geometry.

Abstract

This paper provides an introduction to Kundt spaces, clarifying several important properties, many of which are typically scattered across the mathematical literature or presented without explicit reference to Kundt terminology. While not exhaustive, our approach aims to offer a pedagogical introduction, using a more geometric language and focusing on key concepts directly related to these spaces, such as lightlike totally geodesic foliations.

Introduction to Kundt spaces

TL;DR

Kundt spaces are characterized geometrically by a non-singular lightlike geodesic field whose orthogonal distribution is integrable and totally geodesic. The paper develops a coordinate-free framework, linking twist-free, shear-free, and divergence-free conditions to a transversally Riemannian foliation, and shows these equivalences yield a local metric form that encompasses Brinkmann spaces, pp-waves, plane waves, and Siklos spaces. It classifies examples, analyzes how Lie-group actions and Anosov dynamics generate Kundt structures, and studies global/topological restrictions in low dimensions, including surfaces and 3-manifolds, as well as high-regularity vs low-regularity scenarios. The work also situates Kundt spaces within a hierarchy of Lorentzian geometries and demonstrates that dynamical properties of the isometry group can induce locally Kundt or globally Kundt structures, with implications for both mathematical physics and differential geometry.

Abstract

This paper provides an introduction to Kundt spaces, clarifying several important properties, many of which are typically scattered across the mathematical literature or presented without explicit reference to Kundt terminology. While not exhaustive, our approach aims to offer a pedagogical introduction, using a more geometric language and focusing on key concepts directly related to these spaces, such as lightlike totally geodesic foliations.

Paper Structure

This paper contains 31 sections, 9 theorems, 18 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Some vocabulary
  3. Kundt spaces
  4. Topology and dynamics
  5. Acknowledgment
  6. Generalities: the mathematics behind Kundt spaces
  7. Totally geodesic lightlike foliations
  8. Examples
  9. Actions of Lie groups
  10. Anosov flows and Kundt spaces of low regularity
  11. From Anosov flows to locally Kundt structures
  12. Kundt spaces of low regularity
  13. Local vs Global
  14. Adapted coordinates
  15. Hierarchy
  16. ...and 16 more sections

Key Result

Proposition 2.1

Let $(M,\mathcal{F})$ be a Lorentzian manifold with a codimension $1$ lightlike foliation. Then $\mathcal{F}$ is totally geodesic if and only if its normal foliation $\mathcal{V}$ is leafwise transversally Riemannian, that is, any vector field tangent to $\mathcal{V}$ preserves the degenerate Rieman

Figures (1)

  • Figure 1: A Reeb component on $\mathbb{R}^2$. The foliation induced on $\mathbb{T}^2$ by the action of $\mathbb{Z}^2$ on $\mathbb{R}^2$: $(x,y) \mapsto (x+n,y+m)$ is a Reeb component on $\mathbb{T}^2$

Theorems & Definitions (41)

  • Definition 1.1
  • Definition 1.2
  • Definition 1.3
  • Definition 1.4
  • Definition 1.5
  • Proposition 2.1
  • proof
  • Definition 2.2
  • Definition 2.3
  • Lemma 2.4
  • ...and 31 more