A note on 2-dimensional Finsler manifold

TL;DR

This paper investigates whether a 2-dimensional Finsler manifold with a specific smooth function condition must be Riemannian, addressing a problem that remains open in higher dimensions.

Contribution

It provides a solution to the problem for 2-dimensional Finsler manifolds, clarifying the conditions under which such manifolds are Riemannian.

Findings

For n=2, the condition implies the manifold is Riemannian.

The problem for higher dimensions remains unresolved.

The paper advances understanding of Finsler versus Riemannian structures in low dimensions.

Abstract

We solve the following problem for $n=2:$ Is any n-dimensional Finsler manifold $(M, F)$ with a function $f$ which is nonconstant and smooth on $M$ satisfying $ \dfrac{\partial g^{ij}}{\partial y^k}\dfrac{\partial f}{\partial x^i}=0, $ a Riemannian manifold? The problem for $n>2$ remains open.