On projectively flat Finsler spaces

TL;DR

This paper reviews the history of projectively flat Finsler spaces, provides new criteria and PDE systems for their characterization, and identifies Minkowski spaces as the only parameter-preserving examples over ^n.

Contribution

It introduces a new elementary proof for projective flatness conditions, derives a PDE system for characterization, and characterizes parameter-preserving projectively flat Finsler spaces.

Findings

A simple proof of known projective flatness condition.

A PDE system characterizing projectively flat Finsler spaces.

Minkowski spaces are the only parameter-preserving projectively flat Finsler spaces over ^n.

Abstract

First we present a short overview of the long history of projectively flat Finsler spaces. We give a simple and quite elementary proof of the already known condition for the projective flatness, and we give a criterion for the projective flatness of a special Lagrange space (Theorem 1). After this we obtain a second-order PDE system, whose solvability is necessary and sufficient for a Finsler space to be projectively flat (Theorem 2). We also derive a condition in order that an infinitesimal transformation takes geodesics of a Finsler space into geodesics. This yields a Killing type vector field (Theorem 3). In the last section we present a characterization of the Finsler spaces which are projectively flat in a parameter-preserving manner (Theorem 4), and we show that these spaces over $\mathbb R^n$ are exactly the Minkowski spaces (Theorems 5 and 6).