Berwald metrics constructed by Chevalley's polynomials

TL;DR

This paper explicitly describes Berwald metrics using Chevalley's polynomials, classifies reversible and irreversible cases, and proves their unique determination by Minkowski metrics on maximal flats.

Contribution

It introduces explicit Chevalley polynomial descriptions of Berwald metrics and completes their classification, including reversible and irreversible cases, with new uniqueness results.

Findings

Explicit Chevalley polynomial descriptions of Berwald metrics.

Complete classification of reversible and irreversible Berwald metrics.

Proof that Berwald metrics are uniquely determined by Minkowski metrics on Cartan flats.

Abstract

Berwald metrics are particular Finsler metrics which still have linear Berwald connections. Their complete classification is established in an earlier work, [Sz1], of this author. The main tools in these classification are the Simons-Berger holonomy theorem and the Weyl-group theory. It turnes out that any Berwald metric is a perturbed-Cartesian product of Riemannian, Minkowski, and such non-Riemannian metrics which can be constructed on irreducible symmetric manifolds of $rank > 1$. The existence of these metrics are well established by the above theories. The present paper has several new features. First, the Finsler functions of Berwald manifolds are explicitly described by the Chevalley polynomials. New results are also the complete lists of reversible (d(x,y)=d(y,x)) resp. irreversible ($d(x,y)\not =d(y,x)$) Berwald metrics. The Cartan symmetric Finsler manifolds are also completely determined. The paper is concluded by proving that a Berwald metric is uniquely determined by the Minkowski metric induced on an arbitrarily fixed maximal totalgeodesic flat submanifold (Cartan flat). Moreover, any two Cartan flats are isometric.