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A Toponogov globalisation result for Lorentzian length spaces

Tobias Beran, John Harvey, Lewis Napper, Felix Rott

TL;DR

This work establishes a Lorentzian analogue of Toponogov's Globalisation Theorem for Lorentzian length spaces with lower timelike curvature bounds in the sense of angle comparison. It introduces a robust toolkit—time functions, null distance, Alexandrov-type lemmata, and a Lorentzian cat's cradle—to promote local curvature bounds to global ones, and proves that the entire space becomes a $(\geq K)$-comparison neighbourhood. The results yield synthetic Lorentzian Bonnet–Myers and Splitting theorems under local curvature bounds and demonstrate stability of curvature bounds under Lorentzian Gromov–Hausdorff convergence. These developments broaden the scope of synthetic Lorentzian geometry, providing a bridge between local curvature control and global geometric/topological consequences with potential applications to Lorentzian convergence theory and spacetime models.

Abstract

In the synthetic geometric setting introduced by Kunzinger and Sämann, we present an analogue of Toponogov's Globalisation Theorem which applies to Lorentzian length spaces with lower (timelike) curvature bounds. Our approach utilises a "cat's cradle" construction akin to that which appears in several proofs in the metric setting. On the road to our main result, we also provide a lemma regarding the subdivision of triangles in spaces with a local lower curvature bound and a synthetic Lorentzian version of the Lebesgue Number Lemma. Several properties of time functions and the null distance on globally hyperbolic Lorentzian length spaces are also highlighted. We conclude by presenting several applications of our results, including versions of the Bonnet--Myers Theorem and the Splitting Theorem for Lorentzian length spaces with local lower curvature bounds, as well as discussion of stability of curvature bounds under Gromov--Hausdorff convergence.

A Toponogov globalisation result for Lorentzian length spaces

TL;DR

This work establishes a Lorentzian analogue of Toponogov's Globalisation Theorem for Lorentzian length spaces with lower timelike curvature bounds in the sense of angle comparison. It introduces a robust toolkit—time functions, null distance, Alexandrov-type lemmata, and a Lorentzian cat's cradle—to promote local curvature bounds to global ones, and proves that the entire space becomes a -comparison neighbourhood. The results yield synthetic Lorentzian Bonnet–Myers and Splitting theorems under local curvature bounds and demonstrate stability of curvature bounds under Lorentzian Gromov–Hausdorff convergence. These developments broaden the scope of synthetic Lorentzian geometry, providing a bridge between local curvature control and global geometric/topological consequences with potential applications to Lorentzian convergence theory and spacetime models.

Abstract

In the synthetic geometric setting introduced by Kunzinger and Sämann, we present an analogue of Toponogov's Globalisation Theorem which applies to Lorentzian length spaces with lower (timelike) curvature bounds. Our approach utilises a "cat's cradle" construction akin to that which appears in several proofs in the metric setting. On the road to our main result, we also provide a lemma regarding the subdivision of triangles in spaces with a local lower curvature bound and a synthetic Lorentzian version of the Lebesgue Number Lemma. Several properties of time functions and the null distance on globally hyperbolic Lorentzian length spaces are also highlighted. We conclude by presenting several applications of our results, including versions of the Bonnet--Myers Theorem and the Splitting Theorem for Lorentzian length spaces with local lower curvature bounds, as well as discussion of stability of curvature bounds under Gromov--Hausdorff convergence.
Paper Structure (10 sections, 21 theorems, 19 equations, 7 figures)

This paper contains 10 sections, 21 theorems, 19 equations, 7 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Notation and conventions
  4. Hyperbolic angles and curvature bounds
  5. Null distance
  6. Lorentzian Toponogov Globalisation
  7. Applications and outlook
  8. Gromov--Hausdorff convergence
  9. Geometric consequences
  10. Future work

Key Result

Theorem \ref{thm: LorentzianToponogov}

Let $X$ be a connected, globally hyperbolic, regular Lorentzian length space with a time function $T$ and curvature bounded below by $K\in \mathbb{R}$ in the sense of angle comparison. Then each of the properties in Definition def-cb-ang hold globally; in particular, the entire space $X$ is a $(\geq

Figures (7)

  • Figure 1: A convex situation in the future version of Alexandrov's Lemma.
  • Figure 2: A concave situation in the across version of Alexandrov's Lemma.
  • Figure 3: If the angle condition at $p$ in $\Delta(p,q,r)$ fails to hold (in black), then at least one of the three angles conditions (in red) at $x$ or $p$ in the smaller triangles fail to hold.
  • Figure 4: The angle condition (black) originally fails to hold at $p_0$ in $\Delta(p_0,q_0,r)$. After the first subdivision (dashed), the angle condition (red) fails to hold at $p_1$ in $\Delta(p_1,q_1,r)$. After the second subdivision (dotted), the angle condition (blue) fails to hold at $p_2$ in $\Delta(p_2,q_2,r)$.
  • Figure 5: The cat's cradle construction, showing the first three subtriangles $\Delta_1$, $\Delta_2$ and $\Delta_3$.
  • ...and 2 more figures

Theorems & Definitions (41)

  • Theorem \ref{thm: LorentzianToponogov}
  • Definition 2.1: Regularity
  • Definition 2.2: Comparison angles
  • Lemma 2.3: Law of cosines
  • Corollary 2.4: Law of cosines monotonicity
  • Definition 2.5: Angles
  • Proposition 2.6: Balanced segments in Lo-rentz-ian pre-length space
  • proof
  • Definition 2.7: Curvature bounds by angle comparison
  • Proposition 2.8: Equivalence of curvature bounds
  • ...and 31 more