Solutions for the Landsberg unicorn problem in Finsler geometry

TL;DR

This paper investigates explicit examples of non-Berwaldian Landsberg metrics in Finsler geometry using computational tools, revealing geometric properties and generating new simple examples through deformation techniques.

Contribution

The paper introduces a computational approach with Maple to analyze complex Landsberg metrics, producing new simple examples and insights into their geometric properties.

Findings

Explicit non-Berwaldian Landsberg metrics were constructed.

Deformation of geodesic spray yields new simple examples.

Geometric properties of the spray were characterized.

Abstract

It is still a long-standing open problem in Finsler geometry, is there any regular Landsberg metric which is not Berwaldian. However, there are non-regular Landsberg metrics which are not Berwladian. The known examples are established by G. S. Asanov and Z. Shen. In this paper, we use the Maple program to study some explicit examples of non-Berwaldian Landsberg metrics. In fact, such kinds of examples are very tedious and complicated to investigate. Nonetheless, we use the maple program and Finsler packages to simplify the calculations in an elegant way. Depending on these examples, we manage to figure out some geometric properties of the geodesic spray of a non-Berwaldain Landsberg metric. Deforming this spray in a very specific way, using the metrizability tools of the deformed spray, we get new (very simple) non-Berwaldian Landsberg metrics. Moreover, the powerful of this procedure is investigating a very simple and useful formula for the general class obtained by Z. Shen.