Semi Concurrent vector fields in Finsler geometry

TL;DR

This paper introduces semi-concurrent vector fields in Finsler geometry, showing that their presence often implies the manifold is Riemannian, and explores conditions under which Finsler manifolds are Riemannian or conic Finslerian.

Contribution

It defines semi-concurrent vector fields in Finsler geometry and proves their implications for the Riemannian nature of the manifold, extending Tachibana's characterization.

Findings

Semi-concurrent vector fields imply the manifold is Riemannian in certain cases.

Finsler manifolds with semi-concurrent vector fields are either Riemannian or conic Finslerian.

Conjecture: No regular non-Riemannian Finsler metric admits a semi-concurrent vector field.

Abstract

In the present paper, we introduce and investigate the notion of a semi concurrent vector field on a Finsler manifold. We show that some special Finsler manifolds admitting such vector fields turn out to be Riemannian. We prove that Tachibana's characterization of Finsler manifolds admitting a concurrent vector field leads to Riemannain metrics. We give an answer to the question raised in \cite{DWF}: "Is any n-dimensional Finsler manifold $(M,F)$, admitting a non-constant smooth function $f$ on $M$ such that $\frac{\partial f}{\partial x^i}\frac{\partial g^{ij}}{\partial y^k}=0$, a Riemannian manifold?". Various examples for conic Finsler and Riemannian spaces that admit semi-concurrent vector field are presented. Finally, we conjectured that there is no regular Finsler non-Riemannian metric that admits a semi-concurrent vector field. In other words, a Finsler metric admitting a semi-concurrent vector field is necessarily either Riemannian or conic Finslerian.