Splitting Theorem for Lorentzian Pre-Length Spaces

• The paper establishes that under synthetic curvature bounds, a Lorentzian pre-length space decomposes isometrically as ℝ × S, generalizing classical splitting results.
• It employs techniques such as Busemann functions, flat stripe and rigidity lemmas, and global hyperbolicity to derive the product structure with precise metric and causal criteria.
• The study identifies open problems in achieving full metric-measure splitting under synthetic Ricci bounds, suggesting new avenues in non-smooth Lorentzian geometry.

A splitting theorem for Lorentzian pre-length spaces characterizes when a non-smooth, synthetic Lorentzian space with prescribed curvature bounds and sufficient global completeness properties decomposes isometrically as a product of the real line with a metric space. This theorem generalizes the classical splitting results of Cheeger–Gromoll (for Riemannian metric spaces) and Galloway (for smooth Lorentzian manifolds) to the setting of low-regularity, synthetic Lorentzian geometry, where curvature is defined through triangle or angle comparison rather than differential invariants. It connects synthetic causal geometry, global hyperbolicity, and curvature bounds to rigidity phenomena and product structures.

1. Foundations: Lorentzian Pre-Length and Length Spaces

A Lorentzian pre-length space is a metric-causal quintuple $(X, d, \ll, \le, \tau)$, where $(X, d)$ is a metric space, $\le$ is a causal relation (transitive, closed), $\ll$ is its interior (chronological relation), and $\tau: X \times X \to [0, \infty]$ is a lower semicontinuous time-separation function satisfying:

• Reverse triangle inequality: for $x \le y \le z$, $\tau(x, y) + \tau(y, z) \le \tau(x, z)$.
• $\tau(x, y) > 0$ if and only if $x \ll y$.

A Lorentzian length space further requires the push-up property (causal+chronological relations propagate), causal path-connectedness, local causal closure, and that for each $x \le y$, $\tau(x, y)$ is the supremum of $\tau$-lengths of future-directed causal curves from $x$ to $y$. For completeness, timelike geodesic prolongation demands inextendible geodesics are defined on the whole real line or can be globally extended on larger domains (Kunzinger et al., 2017, Beran et al., 2023, Flores et al., 2024).

2. Synthetic Curvature in Lorentzian Pre-Length Spaces

Synthetic curvature bounds in Lorentzian geometry generalize the notion of sectional and Ricci curvature to metric (even non-smooth) structures. The principal definitions are:

• Lower curvature bounds (triangle/angle comparison): Given $K \in \mathbb{R}$, an open neighborhood $U$ is a $(\ge K)$-comparison neighborhood if each timelike geodesic triangle $\triangle(x,y,z) \subset U$ admits a comparison triangle in the constant curvature Lorentzian model space $\mathbb{L}^2(K)$ with sides matching the $\tau$-lengths. Triangle or angle comparison holds if

$\tau(p,q) \geq \tau_{K}(\bar{p}, \bar{q})$

for all corresponding points on the model triangle (Beran et al., 2023, Kunzinger et al., 2017). The global lower bound is achieved if every point admits such a neighborhood. Angle comparison, in terms of signed upper angles at vertices, is equivalent via the Lorentzian law of cosines (Barton et al., 20 Jan 2026).

• Upper curvature bounds: Analogous upper bounds ($\leq K$) require $\tau(p, q) \leq \tau_K(\bar{p}, \bar{q})$ for points on respective sides.
• Synthetic Ricci-type conditions (weak time-like curvature-dimension, wTCD): For measure-metric Lorentzian spaces, time-like Ricci curvature bounds are encoded via convexity of negative entropy along optimal time-like transport plans, bypassing the need for smooth Ricci tensors (Soultanis, 2023).

3. The Splitting Theorem: Main Results and Hypotheses

The synthetic splitting theorems establish that, under suitable curvature and completeness conditions, the space admits a global isometric decomposition as a Lorentzian product.

Core Statement (Kunzinger–Sämann/BORS, Bartnik Conjecture, and Extensions)

Given a connected, globally hyperbolic Lorentzian length space $(X, d, \ll, \leq, \tau)$ with:

• Global curvature bound: either $(\ge 0)$ or ($\le 0$), i.e., triangle/angle comparison holds everywhere,
• Proper metric,
• Timelike geodesic prolongation/completeness,
• Existence of a complete timelike line $\gamma: \mathbb{R} \to X$,
• In certain cases, topological splitting $X = E \times \mathbb{R}$ with $E$ compact, the following holds: $X \cong \mathbb{R} \times S$ as a Lorentzian product, with time-separation function

$$\tau_{prod}((s,p),(t,q)) = \sqrt{(t-s)^2 - d_S(p,q)^2}\,,$$

where $S$ is a proper, strictly intrinsic CAT(0) or Alexandrov (curvature $\geq 0$) metric length space (Flores et al., 2024, Beran et al., 2023, Barton et al., 20 Jan 2026).

The table summarizes the curvature variants and product structure:

Curvature Bound	Product Structure	Nature of Factor $S$
$\geq 0$	$\mathbb{R} \times S$	Alexandrov curvature $\geq 0$
$\leq 0$	$\mathbb{R} \times S$	CAT(0) metric space

4. Essential Proof Strategy and Technical Tools

The proof architecture is fully synthetic, replacing PDE and smooth geometric tools with metric comparisons, convexity, and stability arguments:

• Busemann functions and level sets: For a complete timelike line $\gamma$, construct forward and backward Busemann functions via limits of time-separation. Convexity and comparison imply level sets of the Busemann function are proper metric spaces of non-negative (Alexandrov) or CAT(0) curvature. The space is foliated by these sets, with each timelike line providing a direction for product decomposition (Kunzinger et al., 2017, Beran et al., 2023).
• Product embedding: The assignment $x \mapsto (b^+(x), \pi(x))$, where $b^+$ is the Busemann function and $\pi(x)$ is projection to a level set, realizes the isometric structure $\mathbb{R} \times S$.
• Flat stripe and rigidity lemmas: In $(\leq 0)$-curvature, uniqueness of parallel lines and the flat strip lemma guarantee the global product, enforcing that all lines parallel to $\gamma$ correspond bijectively to points in $S$ (Barton et al., 20 Jan 2026).
• Toponogov globalisation: To extend local curvature bounds globally, subdivision and glueing arguments (cat’s cradle, Lebesgue number lemma) allow propagation from local to global comparison (Beran et al., 2023).
• Causal boundary and TIP uniqueness: For Bartnik-type settings with compact slices, showing that the future causal boundary consists of a single ideal point is equivalent to global splitting (Flores et al., 2024).
• Measure rigidity (partial splitting): Under synthetic Ricci or energy conditions (e.g., wTCD$_p(0,N)$), one can show partial rigidity, such as constancy of slice-measures, without full metric splitting (Soultanis, 2023).

5. Structural and Measure Rigidity: Full vs. Partial Splitting

In the presence of strong synthetic Ricci bounds or energy conditions, but lacking a well-developed “infinitesimal Hilbertianity” or Bochner-type structures, only partial splitting can be concluded. For Soultanis’s orthogonal product model $hI \times_{\mathcal{F}} X$, entropy convexity in time forces that, under wTCD$(0,N)$, slice measures $m_s$ are independent of the time variable when $h \equiv 1$. This indicates measure rigidity without establishing a global metric product structure (Soultanis, 2023). The full Galloway-type splitting—i.e., isometric decomposition with a Lorentzian product—remains open in this setting.

6. Examples, Corollaries, and Synthesis with Classical Results

• Classical smooth models: For compact Riemannian $E$ of non-negative sectional curvature, the standard ($E \times \mathbb{R}$) Lorentzian product recovers the classical splitting (Flores et al., 2024, Kunzinger et al., 2017).
• Rigidity: Any globally hyperbolic Lorentzian length space with non-negative synthetic curvature, timelike completeness, and a timelike line must split as a Lorentzian product. The result holds even in the absence of differentiable structure (no smooth metric nor CMC hypersurfaces required).
• Comparison with Riemannian and smooth Lorentzian splitting: In the Riemannian setting, lower Ricci bounds and completeness yield Cheeger–Gromoll splitting. In the Lorentzian setting, the analogous result typically requires lower Ricci or sectional curvature bounds, global hyperbolicity, and existence of a timelike line (Galloway theorem). The synthetic results recapitulate these phenomena with curvature notions derived from triangle comparison and weak convexity (Kunzinger et al., 2017, Beran et al., 2023).

7. Open Directions and Outstanding Problems

While the splitting theorem for Lorentzian pre-length spaces is now established under global curvature bounds (sectional-type, both lower and upper), full metric-measure splitting in the presence of only synthetic Ricci bounds or Bakry–Émery-like conditions remains open. The lack of PDE-type tools, infimal Hilbertianity, and control of null-geodesic branching presents technical obstacles (Soultanis, 2023). Developing a theory analogous to the metric-measure splitting in the RCD-framework, but adapted to Lorentzian synthetic geometry, is a current subject of investigation.

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Primary references: (Kunzinger et al., 2017, Beran et al., 2023, Soultanis, 2023, Flores et al., 2024, Barton et al., 20 Jan 2026).