A Splitting Theorem for non-positively curved Lorentzian spaces

We prove a splitting theorem for Lorentzian pre-length spaces with global non-positive timelike curvature. Additionally, we extend the first variation formula to spaces with any timelike curvature bound, either from above or below, and different from 0.

💡 Research Summary

This paper establishes a splitting theorem for Lorentzian pre‑length spaces whose timelike curvature is globally bounded above by a non‑positive constant, and it also extends the first variation formula to spaces with arbitrary timelike curvature bounds, both from above and below.

The authors begin by reviewing the synthetic framework of Lorentzian pre‑length spaces (X, d, ≤, ≪, τ), where d is a metric, ≤ and ≪ are causal and chronological relations, and τ is the time‑separation function. They define timelike triangles, comparison triangles in the model spaces L₂(K) of constant curvature K, and introduce (≥K)‑ and (≤K)‑comparison neighbourhoods. Within a (≤K)‑comparison neighbourhood geodesics are unique, while in a (≥K)‑neighbourhood comparison angles are finite. Angles between geodesics are defined via limits of comparison angles, and a signed version is introduced to handle opposite time orientations. Basic triangle inequalities for angles and an angle‑sum formula in Minkowski space are proved.

The first major technical contribution is a first variation formula that works for any curvature bound K. After establishing the formula in the model spaces (Proposition 3.1), the authors prove a general version (Theorem 3.6) for arbitrary Lorentzian pre‑length spaces with timelike curvature bounded either above or below. This extends previous results, which were limited to non‑negative lower bounds.

Next, the paper proves a rigidity statement for upper curvature bounds (Proposition 4.1): if equality holds in the comparison inequality for a triangle in a (≤K)‑neighbourhood, then the triangle is isometric to its model triangle in L₂(K). This rigidity is crucial for the later construction of parallel lines and rays.

In Section 5 the authors study parallelism. Given a complete timelike line γ and a point p not on γ, they construct two parallel rays—one future‑directed and one past‑directed—starting at p and asymptotic to γ. They show that the angle between these two rays is zero precisely when the rays concatenate to a full timelike line; otherwise the angle is positive, preventing concatenation. This phenomenon explains why spaces with positive upper curvature bounds (e.g., de Sitter space) do not split as a product.

The main result, Theorem 6.6, combines the previous ingredients. Assume X is a Lorentzian pre‑length space with global timelike curvature ≤ 0 and that X contains a complete timelike line γ. Let S be the set of all complete timelike lines parallel to γ. The authors endow S with a metric that makes it a CAT(0) space. Then the map
 ℝ × S → X, (t, α) ↦ α(t)
is an isometry of Lorentzian pre‑length spaces, i.e., X splits isometrically as the Lorentzian product ℝ × S. Moreover, if X itself equals the union of the parallel lines (i.e., X = ⋃_{α∈S} α(ℝ)), the splitting is global.

The paper concludes by emphasizing that, unlike the classical Riemannian case where lower curvature bounds are essential for splitting, in the Lorentzian synthetic setting an upper curvature bound ≤ 0 suffices, provided a complete timelike line exists. This opens new avenues for the study of non‑smooth Lorentzian spaces, their causal structure, and applications to general relativity and optimal transport.