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Transverse Geometry of Lorentzian foliations with applications to Lorentzian orbifolds

Francisco C. Caramello, Henrique A. Puel Martins, Ivan P. Costa e Silva

Abstract

We prove a transverse diameter theorem in the context of Lorentzian foliations, which can be interpreted as a Hawking--Penrose-type singularity theorem for timelike geodesics transverse to the foliation. In order to develop the necessary machinery we introduce and study a novel causality structure on the leaf space via the transverse Lorentzian geometry on the foliated manifold. We describe the initial rungs of a transverse causal ladder and relate them to their standard counterparts on an underlying foliated spacetime. We show how these results can be interpreted as doing Lorentzian (and more generally semi-Riemannian) geometry on low-regularity spaces that can be realized as leaf spaces of foliations. Accordingly, we discuss how all of these concepts and results apply to Lorentzian orbifolds, insofar as these can be seen as leaf spaces of a specific class of Lorentzian foliations. In particular, we derive an associated Lorentzian timelike diameter theorem on orbifolds.

Transverse Geometry of Lorentzian foliations with applications to Lorentzian orbifolds

Abstract

We prove a transverse diameter theorem in the context of Lorentzian foliations, which can be interpreted as a Hawking--Penrose-type singularity theorem for timelike geodesics transverse to the foliation. In order to develop the necessary machinery we introduce and study a novel causality structure on the leaf space via the transverse Lorentzian geometry on the foliated manifold. We describe the initial rungs of a transverse causal ladder and relate them to their standard counterparts on an underlying foliated spacetime. We show how these results can be interpreted as doing Lorentzian (and more generally semi-Riemannian) geometry on low-regularity spaces that can be realized as leaf spaces of foliations. Accordingly, we discuss how all of these concepts and results apply to Lorentzian orbifolds, insofar as these can be seen as leaf spaces of a specific class of Lorentzian foliations. In particular, we derive an associated Lorentzian timelike diameter theorem on orbifolds.
Paper Structure (15 sections, 46 theorems, 114 equations)

This paper contains 15 sections, 46 theorems, 114 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Metrics on vector bundles
  4. Basics of foliation theory
  5. Holonomy
  6. Orbifolds
  7. Transverse structures
  8. Semi-Riemannian foliations
  9. Elements of transverse causality
  10. The transverse causal ladder
  11. Transverse Lorentzian causality versus standard causality
  12. A transverse diameter theorem
  13. Contrasting Ricci and transverse Ricci bounds
  14. Transverse diameter
  15. Lorentzian orbifold causality

Key Result

Theorem 1

Let $(M,\mathcal{F}, g_\intercal)$ be a codimension $q$ transversely globally hyperbolic Lorentzian foliation with compact leaves and such that the leaf space $M/\mathcal{F}$ is Hausdorff. If the transverse Ricci tensor obeys the inequality $\mathop{\mathrm{Ric}}\nolimits_\intercal\geq (q-1)C>0$ on

Theorems & Definitions (123)

  • Theorem 1
  • Corollary 2
  • Theorem 3
  • Example 2.1: Submersions $\&$ Bundles
  • Example 2.2: Homogeneous foliations
  • Example 2.3: Integrable distributions
  • Example 2.4: Suspensions
  • Example 2.5: Pullbacks
  • Example 2.6
  • Proposition 2.7
  • ...and 113 more