Fibrewise Real-Analytic Landsberg Metrics in Dimension at Least Four Are Berwald
Abstract
Let M be a smooth manifold of dimension n≥4, and let F be a smooth, strongly convex Landsberg metric, regular on the entire slit tangent bundle. We prove that F is Berwald if, at each fixed base point, the Minkowski norm is real-analytic on the entire slit tangent space. The proof rests on two analytic rigidity theorems for a single Minkowski space. The first asserts that, in dimension at least $3$, a real-analytic Minkowski norm whose Hessian metric is flat is induced by an inner product. The second asserts that, in dimension at least $4$, the Hessian metric of a real-analytic Minkowski norm is flat if the entire indicatrix carries a nonzero smooth symmetric quartic form satisfying a twisted Codazzi equation. The quartic form need not be real-analytic, and the second theorem is proved independently of the first. Applying these results to the angular Berwald quartic form shows that the metric is Riemannian on the open set of base points on whose fibres the Berwald curvature is not identically zero. Since every Riemannian metric is Berwald, this set must be empty. No analytic structure, compactness or completeness of the base manifold is assumed, and the metric need not be reversible.
Related
- Schwarz lemma on bounded symmetric domains endowed with holomorphic invariant Kähler--Berwald metrics
- The Rigidity of the closed three dimensional regular Landsberg metrics
- Non-comparable invariant metrics and failure of Gromov hyperbolicity on a real-analytic pseudoconvex domain
- A Model-threshold Dimension Bound and Sharp Critical Ends for Smooth Singular Sets of Constant Positive σk-curvature Metrics
- Dimension Thresholds for Relative-Error Representations of Finite Metrics in Pseudo-Euclidean Spaces
BibTeX
@misc{li2026fibrewise,
title = {{Fibrewise Real-Analytic Landsberg Metrics in Dimension at Least Four Are Berwald}},
author = {Ming Li and Kefeng Liu},
howpublished = {arXiv},
year = {2026},
month = nov,
doi = {10.48550/arxiv.2611.05067},
eprint = {2611.05067},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2611.05067},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Fibrewise Real-Analytic Landsberg Metrics in Dimension at Least Four Are Berwald
AU - Ming Li
AU - Kefeng Liu
PY - 2026
DA - 2026/11/03
PB - arXiv
DO - 10.48550/arxiv.2611.05067
UR - https://arxiv.org/abs/2611.05067
AB - Let M be a smooth manifold of dimension n≥4, and let F be a smooth, strongly convex Landsberg metric, regular on the entire slit tangent bundle. We prove that F is Berwald if, at each fixed base point, the Minkowski norm is real-analytic on the entire slit tangent space. The proof rests on two analytic rigidity theorems for a single Minkowski space. The first asserts that, in dimension at least $3$, a real-analytic Minkowski norm whose Hessian metric is flat is induced by an inner product. The second asserts that, in dimension at least $4$, the Hessian metric of a real-analytic Minkowski norm is flat if the entire indicatrix carries a nonzero smooth symmetric quartic form satisfying a twisted Codazzi equation. The quartic form need not be real-analytic, and the second theorem is proved independently of the first. Applying these results to the angular Berwald quartic form shows that the metric is Riemannian on the open set of base points on whose fibres the Berwald curvature is not identically zero. Since every Riemannian metric is Berwald, this set must be empty. No analytic structure, compactness or completeness of the base manifold is assumed, and the metric need not be reversible.
ER -