On the generalization of theorems from Riemannian to Finsler Geometry I: Metric Theorems

TL;DR

This paper introduces a method to extend metric-based theorems from Riemannian to Finsler geometry, demonstrating their equivalence in topology, providing a new proof of Hopf-Rinow, and establishing the existence of a Finslerian center of mass.

Contribution

It presents a novel approach for generalizing metric theorems from Riemannian to Finsler geometry, including key topological and geometric results.

Findings

Finsler topology is equivalent to manifold topology

New proof of the Hopf-Rinow theorem in Finsler geometry

Existence of the center of mass in non-reversible Finsler spaces

Abstract

A method to generalize results from Riemannian Geometry to Finsler geometry is presented. We use the method to generalize several results that involve only metric conditions. Between them we show that the topology induced by the Finsler structure is equivalent to the manifold topology, we provide a new proof of the Hopf-Rinow theorem in Finsler geometry and we prove the existence of the center of mass of a convex body when the non-symmetric distance function comes from a non-reversible Finsler function.