The Landsberg equation of a Finsler space

TL;DR

This paper introduces the Landsberg equation for Finsler spaces and proves that certain Landsberg metrics must be Berwald, resolving a long-standing open problem in Finsler geometry within a broad class of metrics.

Contribution

It establishes that $(oldsymbol{ ext{$oldsymbol{ ext{(}}oldsymbol{ ext{$oldsymbol{ ext{}} ext{}}}$oldsymbol{ ext{)}}}$-metrics of Landsberg type are necessarily Berwald, showing the non-existence of Landsberg unicorns in this class.

Findings

Landsberg equation is introduced for Finsler spaces.

Landsberg metrics in the $( ext{$ ext{α}_1, ext{α}_2$})$-class are shown to be Berwald.

The long-standing Landsberg unicorn problem is resolved for this broad class.

Abstract

Given a Finsler space, we introduce a system of partial differential equations, called the Landsberg equation. Based on a careful analysis of the Landsberg equation and the observation that the solution space is invariant under the linear isometries of the tangent Minkowski spaces, we prove that an $(\alpha_1, \alpha_2)$-metric of the Landsberg type must be a Berwald metric. This shows that the hunting for a unicorn, one of the longest standing open problem in Finsler geometry, cannot be successful even in the very broad class of $(\alpha_1,\alpha_2)$-metrics.