Some fundamental problems in global Finsler geometry

Abstract

The geometry and analysis on Finsler manifolds is a very important part of Finsler geometry. In this survey 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. Further, we also obtain an estimate of lower bound for the non-zero eigenvalues of the Finsler Laplacian under the condition that $\mathrm{Ric}_{N}\geq K >0 $.

Keywords

• dual finsler metric
• gradient vector field
• finsler laplacian
• eigenvalue
• hessian
• lie derivative
• weighted ricci curvature