Deformations and Hilbert's Fourth Problem

TL;DR

This paper classifies a specific class of Finsler metrics, showing that their projective flatness in higher dimensions is derived from the flatness of an underlying Riemannian metric, through a special deformation process.

Contribution

It provides a classification of projectively flat Finsler metrics defined by Riemannian metrics and 1-forms, revealing their flatness originates from Riemannian geometry.

Findings

Projectively flat Finsler metrics are characterized by special deformations.

Flatness of these Finsler metrics stems from the flatness of associated Riemannian metrics.

The classification applies to dimensions n ≥ 3.

Abstract

In this paper we study a class of Finsler metrics defined by a Riemannian metric and an 1-form. We classify those of projectively flat in dimension $n\geq3$ by a special class of deformations. The results show that the projective flatness of such kind of Finsler metrics always arises from that of some Riemannian metric.