On the Berwald-Landsberg problem

TL;DR

This paper investigates the geometric properties of Finsler spaces by explicitly computing average Riemannian metrics and demonstrates that regular Landsberg spaces are necessarily Berwald spaces, advancing the understanding of Finsler geometry.

Contribution

It explicitly determines the Levi-Civita connection of average metrics and proves that regular Landsberg spaces are Berwald spaces, resolving a key problem in Finsler geometry.

Findings

Explicit Levi-Civita connection for average metrics

Invariance of the average metric under homotopy

Regular Landsberg spaces are Berwald spaces

Abstract

Given a Finsler space (M,F), one can define natural average Riemannian metrics on M by averaging on the indicatrix I_x the fundamental tensor g of the Finsler function $F$. In this paper we determine explicitly the Levi-Civita connection for these average Riemannian metrics. We apply the result to the case when (M,F) is a Landsberg space. Using a particular averaging procedure, the invariance of the average metric along a homotopy in the space of Finsler structures over M is shown. As a consequence of such invariance, we prove that any C^5 regular Landsberg space is a Berwald space.