[Paper Review] Berwald metrics constructed by Chevalley's polynomials

Zoltán Szabó|ArXiv.org|Jan 21, 2006

Advanced Differential Geometry Research20 references21 citations

TL;DR

This paper provides an explicit construction of Berwald metrics using Chevalley polynomials, classifying all reversible and irreversible Berwald metrics and fully determining Cartan symmetric Finsler manifolds. The key contribution is proving that a Berwald metric is uniquely determined by its Minkowski norm on any fixed maximal totally geodesic flat (Cartan flat), with all such flats being isometric.

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 ($d(x,y) ot =d(y,x)$) 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.

Motivation & Objective

• To provide an explicit description of Finsler functions for Berwald metrics using Chevalley polynomials.
• To complete the classification of reversible and irreversible Berwald metrics.
• To fully determine the structure of Cartan symmetric Finsler manifolds.
• To establish that a Berwald metric is uniquely determined by its restriction to a single Cartan flat.
• To prove that any two Cartan flats in a Berwald manifold are isometric.

Proposed method

• Utilizes Chevalley polynomials to explicitly construct Finsler functions on tangent spaces of Berwald manifolds.
• Applies the Simons-Berger holonomy theorem to identify Riemannian connections that admit non-Riemannian Berwald extensions.
• Employs Weyl group invariance on Cartan subalgebras to characterize holonomy-invariant Finsler norms.
• Uses de Rham decomposition to extend results from irreducible to reducible symmetric spaces.
• Constructs Berwald metrics as parallel extensions of Weyl-invariant norms from Cartan subalgebras to the full tangent space.
• Establishes uniqueness of Berwald metrics via isometry of Cartan flats and Minkowski structure on them.

Experimental results

Research questions

• RQ1 How can Finsler functions of Berwald metrics be explicitly described using Chevalley polynomials?
• RQ2 What are the complete lists of reversible and irreversible Berwald metrics?
• RQ3 Which Cartan symmetric Finsler manifolds exist, and how are they characterized?
• RQ4 To what extent is a Berwald metric determined by its Minkowski norm on a single Cartan flat?
• RQ5 Are all Cartan flats in a Berwald manifold isometric, and how does this relate to global metric uniqueness?

Key findings

• Berwald metrics are completely classified as perturbed-Cartesian products of Riemannian, Minkowski, and non-Riemannian metrics on irreducible symmetric spaces of rank > 1.
• The Finsler functions of all Berwald metrics are explicitly constructed via Chevalley polynomials on Cartan subalgebras.
• Reversible and irreversible Berwald metrics are fully listed, with the latter arising from non-symmetric irreducible factors in the de Rham decomposition.
• Cartan symmetric Finsler manifolds are completely determined by the Weyl group invariance of their Minkowski norms on Cartan subalgebras.
• A Berwald metric is uniquely determined by the Minkowski metric induced on any fixed maximal totally geodesic flat (Cartan flat).
• Any two Cartan flats in a Berwald manifold are locally isometric, confirming the global uniqueness of the metric structure.