Classification of complete Finsler manifolds through a second order differential equation

TL;DR

This paper introduces a second order differential equation approach to classify complete Finsler manifolds, expanding the understanding of their geometry and providing new examples with positive constant sectional curvature.

Contribution

It defines adapted coordinates on Finsler manifolds using a second order differential equation and generalizes Riemannian results to Finsler geometry.

Findings

Classified complete Finsler manifolds using a second order differential equation.

Constructed examples of Finsler metrics with positive constant sectional curvature.

Extended Riemannian geometric results to the Finsler setting.

Abstract

By using a certain second order differential equation, the notion of adapted coordinates on Finsler manifolds is defined and some classifications of complete Finsler manifolds are found. Some examples of Finsler metrics, with positive constant sectional curvature, not necessarily of Randers type nor projectively flat, are found. This work generalizes some results in Riemannian geometry and open up, a vast area of research on Finsler geometry.