A Toponogov globalisation result for Lorentzian length spaces

Area of Science:

• Differential Geometry
• General Relativity
• Metric Geometry

Background:

• Lorentzian length spaces are a generalization of Lorentzian manifolds.
• Curvature bounds are crucial for understanding the geometry of these spaces.
• Toponogov's Globalisation Theorem is a fundamental result in Riemannian geometry.

Purpose of the Study:

• To present a synthetic analogue of Toponogov's Globalisation Theorem for Lorentzian length spaces.
• To extend results concerning curvature bounds in Lorentzian geometry.
• To explore applications in general relativity and geometric analysis.

Main Methods:

• Utilizing a
• Meta_Description='Explore a new Globalisation Theorem for Lorentzian length spaces with timelike curvature bounds, with applications in geometry and relativity.'

Main Results:

• A synthetic analogue of Toponogov's Globalisation Theorem for Lorentzian length spaces with lower timelike curvature bounds.
• A lemma on triangle subdivision and a synthetic Lorentzian Lebesgue Number Lemma.
• Properties of time functions and null distance on globally hyperbolic Lorentzian length spaces.

Conclusions:

• The presented results extend classical geometric theorems to the synthetic Lorentzian setting.
• Applications include versions of the Bonnet-Myers and Splitting Theorems.
• Stability of curvature bounds under Gromov-Hausdorff convergence is discussed.