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Space of Timelike Directions and Curvature Bounds

Joe Barton, Jona Röhrig

Abstract

We investigate the consequences of timelike sectional curvature bounds in Lorentzian length spaces for the existence and structure of the space of directions at a point. It is established that, under upper timelike sectional curvature bounds, the space of directions exists and is itself a metric space with curvature bounded above by $-1$. Furthermore, the metric cone over the space of directions, which canonically models the tangent space at a given point, is shown to constitute a Lorentzian length space with timelike sectional curvature bounded above by $0$. To do this, we introduce the notion of $ε$-$μ$ timelike sectional curvature bounds, which are compatible with pre-existing synthetic curvature conditions. These results extend the comparison-geometric framework to the Lorentzian setting, providing a synthetic characterization of geodesics, tangent cones, and curvature under causal constraints.

Space of Timelike Directions and Curvature Bounds

Abstract

We investigate the consequences of timelike sectional curvature bounds in Lorentzian length spaces for the existence and structure of the space of directions at a point. It is established that, under upper timelike sectional curvature bounds, the space of directions exists and is itself a metric space with curvature bounded above by . Furthermore, the metric cone over the space of directions, which canonically models the tangent space at a given point, is shown to constitute a Lorentzian length space with timelike sectional curvature bounded above by . To do this, we introduce the notion of - timelike sectional curvature bounds, which are compatible with pre-existing synthetic curvature conditions. These results extend the comparison-geometric framework to the Lorentzian setting, providing a synthetic characterization of geodesics, tangent cones, and curvature under causal constraints.
Paper Structure (10 sections, 9 theorems, 57 equations)

This paper contains 10 sections, 9 theorems, 57 equations.

Table of Contents

  1. Introduction
  2. Sectional Curvature Bounds
  3. Previous notions of curvature bounds
  4. epsilon-mu sectional curvature bounds
  5. The Space of Directions
  6. Definitions
  7. Under Upper Curvature Bounds the Space of Directions is a Length Space
  8. Upper Curvature Bounds for the Space of Directions
  9. Curvature Bounds of the Tangent Cone
  10. Curvature bounds for the space of direction

Key Result

Lemma 2.10

The following conditions are equivalent for a strongly causal and regular Lorentzian pre-length space $(X,d,\ll,\leq,\tau)$ :

Theorems & Definitions (39)

  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Definition 2.4
  • Remark 1
  • Definition 2.5
  • Remark 2
  • Definition 2.6
  • Remark 3
  • Remark 4
  • ...and 29 more