Some fundamental problems in global Finsler geometry

TL;DR

This paper explores fundamental issues in global Finsler geometry, refining definitions and characterizations of key concepts like Lie derivatives and gradient fields on Finsler and Randers manifolds.

Contribution

It introduces important topics in global Finsler geometry, optimizes Lie derivative definitions, and characterizes gradient vector fields with new estimates.

Findings

Optimized definitions of Lie derivatives on Finsler manifolds

Characterized gradient vector fields on Randers manifolds

Established gradient estimates for smooth functions

Abstract

The geometry and analysis on Finsler manifolds is a very important part of Finsler geometry. In this article, we introduce some important and fundamental topics in global Finsler geometry and discuss the related properties and the relationships in them. In particular, we optimize and improve the various definitions of Lie derivatives on Finsler manifolds. We also characterize the gradient vector fields and obtain a gradient estimate for any smooth function on a Randers manifold.