A note on 2-dimensional Finsler manifold

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 $\frac{\partial g^{ij}}{\partial y^k}\frac{\partial f}{\partial x^i}=0,$ a Riemannian manifold? The problem for $n>2$ remains open.

arXiv commentarXiv admin note: text overlap with arXiv:1110.6826 by other authors