A new look at Finsler surfaces and the Landsberg's PDE

TL;DR

This paper introduces a novel approach to Finsler surfaces, reduces Landsberg's PDE to a single nonlinear equation, and finds solutions that are conformally Berwaldain, advancing understanding of Landsberg surfaces.

Contribution

The paper simplifies Landsberg's PDE to a single nonlinear PDE and provides explicit solutions using a new perspective on Finsler surfaces.

Findings

Reduced Landsberg's PDE to a single nonlinear PDE

Found a class of solutions to Landsberg's PDE

Showed solutions are conformally Berwaldain

Abstract

In this paper, we introduce a new look at Finsler surfaces. Landsberg surfaces are Finsler surfaces that are solutions of a system of non-linear partial differential equations. Considering the unicorn's Landsberg problem, we reduce this system to a single non-linear PDE which we call the Landsberg's PDE. By making use of the new look of Finsler surfaces, we solve the Landsberg's PDE and get a class of solutions. Moreover, we show that these solutions and their conformal transformations are Berwaldain.