On Berwald Spaces with non-Zero Flag Curvature
A. Tayebi, B. Najafi
TL;DR
This paper establishes strong rigidity for Berwald Finsler manifolds under curvature constraints: if the flag curvature ${\bf K}$ is nowhere zero, the metric must be Riemannian; positively curved constant isotropic Berwald metrics are forced to be Riemannian or locally Minkowskian; compact isotropic Berwald manifolds with strict sign of curvature are Berwald; and homogeneous isotropic Berwald metrics fall into a short list of possibilities, including Randers-type Berwald metrics. The results extend classical rigidity theorems of Numata and Szabó and connect Berwald curvature, S-curvature, and Cartan torsions to global geometric conclusions. The proofs hinge on intricate relations among $\mathbf{B}$, $\mathbf{C}$, $\mathbf{I}$, $\mathbf{J}$, $\mathbf{S}$, and the Riemann curvature, together with curvature bounds and the structure of isotropic Berwald metrics. Collectively, these findings sharpen the landscape of homogeneous and isotropic Berwald geometries, clarifying when non-Riemannian examples can occur and when rigidity enforces Riemannian or Minkowskian models.
Abstract
We prove that every Berwald manifold with non-zero flag curvature is Riemannian. This result provides an extension of Numata and Szabo's rigidity theorems. We show that every positively curved constant isotropic Berwald manifold is Riemannian or locally Minkowskian. Then, we prove that every compact strictly positive (or negative) isotropic Berwald manifold reduces to a Berwald manifold. Finally, we prove that every homogeneous isotropic Berwald metric is either locally Minkowskian, or Riemannian, or Berwald metric or Berwald-Randers metric generalizing result previously only known in the case of Randers metric.