What the study found
The authors prove a synthetic Lorentzian Cartan-Hadamard theorem. In the setting of Lorentzian geometry, a timelike geodesic is the shortest path followed by an object moving slower than light; their result gives existence and uniqueness of timelike geodesics between any pair of timelike related points under additional assumptions.
Why the authors say this matters
The study suggests this result transfers the corresponding statement for locally convex metric spaces to the Lorentzian setting and generalizes the smooth Lorentzian theorem to synthetic Lorentzian geometry. The authors also say it provides a globalization result for their notion of concavity and an application to non-negative upper timelike curvature bounds.
What the researchers tested
The researchers developed an appropriate notion of local concavity for Lorentzian (pre-)length spaces. They worked within synthetic Lorentzian geometry and Lorentzian length spaces, and they assumed global hyperbolicity and future one-connectedness for the geodesic result.
What worked and what didn't
Their approach allowed them to establish existence and uniqueness of timelike geodesics between timelike related points under the stated assumptions. They also prove a globalization result for concavity in Lorentzian length spaces and derive a globalization statement for non-negative upper timelike curvature bounds.
What to keep in mind
The abstract states that the geodesic conclusion requires global hyperbolicity and future one-connectedness. It does not describe further limitations, examples, or cases where the result fails.
Key points
- The paper proves a synthetic Lorentzian Cartan-Hadamard theorem.
- It establishes existence and uniqueness of timelike geodesics between timelike related points under added assumptions.
- The framework uses local concavity for Lorentzian (pre-)length spaces.
- The authors say the result transfers a metric-space statement and generalizes a smooth Lorentzian theorem.
- They also provide a globalization result for concavity and an application to non-negative upper timelike curvature bounds.
Disclosure
- Research title:
- Synthetic Lorentzian Cartan-Hadamard theorem established
- Authors:
- Darius Erös, Sebastian Gieger
- Publication date:
- 2026-04-22
- DOI:
- 10.1090/btran/248
- OpenAlex record:
- View
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