An equivalence theorem of a class of Minkowski norms and its applications

TL;DR

This paper investigates the Cartan tensors of $(eta)$-norms, proves an equivalence theorem for these norms, and shows that certain $(eta)$-metrics with specific curvature properties are Berwald manifolds in Finsler geometry.

Contribution

It establishes an equivalence theorem for $(eta)$-norms and applies it to characterize certain Finsler metrics as Berwald manifolds.

Findings

Cartan tensors of $(eta)$-norms are analyzed in detail

An equivalence theorem for $(eta)$-norms is proved

Vanishing Landsberg curvature implies the metric is Berwald in certain dimensions

Abstract

In this paper, the Cartan tensors of the $(\alpha,\beta)$-norms are investigated in details. Then an equivalence theorem of $(\alpha,\beta)$-norms is proved. As a consequence in Finsler geometry, general $(\alpha,\beta)$-metrics on smooth manifolds of dimension $n\geq4$ with vanishing Landsberg curvatures must be Berwald manifolds.