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arXiv · 2603.06458

Space of Timelike Directions and Curvature Bounds

Joe Barton·Jona Röhrig
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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 $\epsilon$-$\mu$ 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.

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Joe Barton, Jona Röhrig. 2026-03-06. Space of Timelike Directions and Curvature Bounds. https://arxiv.org/abs/2603.06458

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