On the problem of non Berwaldian Landsberg spaces

TL;DR

This paper investigates the existence of non-Berwaldian Landsberg spaces through conformal transformations, providing conditions, examples, and cases where such spaces cannot be Berwaldian, thus advancing understanding in Finsler geometry.

Contribution

It introduces a conformal transformation approach to study Landsberg spaces, derives conditions for non-Berwaldian cases, and provides explicit examples and classifications.

Findings

Existence of Landsberg spaces with non-vanishing T-tensor under conformal transformation

Necessary conditions for Landsberg spaces to be Berwaldian

Examples of non-Berwaldian Landsberg spaces, including singular cases

Abstract

In this paper, we study the long existence problem of non Berwaldian Landsberg spaces using the conformal transformation point of view. Under conformal transformation, the Berwald and Landesberg tensors are calculated in terms of the T-tensor. By giving examples, we show that under conformal transformation, there are Landsberg spaces with non-vanishing T-tensor. A necessary condition for a Landsberg space to be Berwaldian is given. Various special cases are studied. Cases in which the Landsberg spaces can not be Berwaldian are shown. Examples of non-Berwaldian landsberg (singular) spaces are given.