Landsberg–Berwald conjecture
For every smooth, strongly convex, regular Finsler metric on a smooth manifold , if is a Landsberg metric, then is a Berwald metric.
References
Primary source
Additional references
- The Rigidity of the closed three dimensional regular Landsberg metrics — arXiv — Jianyu Mao, Linfeng Zhou
Progress summary
A new unrefereed paper claims the conjecture in closed three-dimensional spaces, but the full question about regular metrics remains open.
The conjecture asks whether every regular Landsberg metric must be Berwald. Szabó announced a proof, but a 2008 analysis identified a gap in the argument.
Known results
- Szabó's claimed proof was found to confuse derivatives in the base and fiber variables (2008).
- Non-Berwaldian examples due to Asanov and Shen are non-regular and therefore do not settle the conjecture (2019).
- Asanov's Finsleroid-Finsler examples are singular or -local, not regular and -global (2016).
- New explicit non-regular examples in dimensions likewise do not settle the regular problem (2026).
September 10, 2026 three-dimensional claim
Jianyu Mao and Linfeng Zhou's article The Rigidity of the closed three dimensional regular Landsberg metrics claims the no-unicorns assertion for closed three-manifolds, without a reversibility assumption. This is a substantial partial result, but it is unrefereed and its verification was not found.
Current status (as of September 2026): The three-dimensional closed-manifold case is claimed, but unverified; the general regular Landsberg--Berwald conjecture remains open.
Solutions 0
No solutions have been posted yet.