The Rigidity of the closed three dimensional regular Landsberg metrics
Abstract
The Landsberg Berwald conjecture asks whether every regular Landsberg metric is Berwald. We settle this conjecture for closed three-dimensional manifolds: every smooth strongly convex regular Landsberg metric on such a manifold is Berwald, without any reversibility assumption. Equivalently, there are no regular Landsberg "unicorns" on closed three-manifolds; this is a reformulation of the same conjecture, rather than a second independent conjecture.The proof combines a vanishing theorem for commuting Codazzi cubic tensors on closed surfaces, a rigidity theorem for three dimensional Minkowski norms with constant curvature indicatrices, and a rank one argument for the nonlinear curvature. The remaining R-quadratic case is settled by compactness along the geodesic flow. The fibrewise constant curvature theorem is an ingredient in this argument and does not assert the full Laugwitz conjecture.
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BibTeX
@misc{mao2026rigidity,
title = {{The Rigidity of the closed three dimensional regular Landsberg metrics}},
author = {Jianyu Mao and Linfeng Zhou},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.11718},
eprint = {2609.11718},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.11718},
note = {CC BY-NC-ND 4.0},
}
RIS
TY - UNPB
TI - The Rigidity of the closed three dimensional regular Landsberg metrics
AU - Jianyu Mao
AU - Linfeng Zhou
PY - 2026
DA - 2026/09/10
PB - arXiv
DO - 10.48550/arxiv.2609.11718
UR - https://arxiv.org/abs/2609.11718
AB - The Landsberg Berwald conjecture asks whether every regular Landsberg metric is Berwald. We settle this conjecture for closed three-dimensional manifolds: every smooth strongly convex regular Landsberg metric on such a manifold is Berwald, without any reversibility assumption. Equivalently, there are no regular Landsberg "unicorns" on closed three-manifolds; this is a reformulation of the same conjecture, rather than a second independent conjecture.The proof combines a vanishing theorem for commuting Codazzi cubic tensors on closed surfaces, a rigidity theorem for three dimensional Minkowski norms with constant curvature indicatrices, and a rank one argument for the nonlinear curvature. The remaining R-quadratic case is settled by compactness along the geodesic flow. The fibrewise constant curvature theorem is an ingredient in this argument and does not assert the full Laugwitz conjecture.
ER -