Solutions for the Landsberg unicorn problem in Finsler geometry

摘要

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 Berwaldian. 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 calculations. Depending on these examples, we manage to figure out some geometric properties of the geodesic spray of a non-Berwaldian 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 power of this procedure consists in investigating a simple and useful formula for the general class obtained by Z. Shen. (c) 2020 Elsevier B.V. All rights reserved.