In some weird spaces, the 'shortest path' between two points can actually split and head in two different directions at the same time.
March 25, 2026
Original Paper
Interior singularity and branching of geodesics in real-analytic sub-Riemannian manifolds
arXiv · 2603.23068
The Takeaway
In standard geometry, the shortest route between two points is a unique line. This research reveals that in certain complex mathematical spaces, two shortest paths can overlap perfectly for a while and then suddenly split into separate directions, defying our basic intuition of how distance and movement work.
From the abstract
We study the regularity and branching of strictly abnormal minimizing geodesics in sub-Riemannian geometry. We construct examples of real-analytic sub-Riemannian manifolds admitting minimizing geodesics that lose regularity at an interior point of their domain and exhibit branching, thereby resolving longstanding open questions.