A comparative overview of Riemannian and Finsler geometry

Abstract

The aim of this article is to present a comparative overview of Riemannian and Finsler geometry, starting from some historical developments to the various directions of current research. This includes the discussion on classification of Finsler spaces of constant flag curvature and constant Ricci curvature, cut locus, conjugate locus, Comparison theorems, Gauss-Bonnet-Chern Theorem and sphere theorem in Finsler geometry. The topological, differential and metric structures on Riemannian manifolds in the presence of convex functions have been active fields of research in the second half of the last century. We discuss some of these results on Riemannian manifolds with convex functions and their recently extended analogues on Finsler manifolds. © 2025 American Mathematical Society.