On the classification of projectively flat Finsler metrics with constant flag curvature

TL;DR

This paper classifies locally projectively flat Finsler metrics with constant flag curvature by solving nonlinear PDEs, providing algebraic classifications for different curvatures and constructing new examples, simplifying the classification of spherical symmetric metrics.

Contribution

It offers a complete algebraic classification of such metrics for all constant curvatures and introduces a new explicit example with flag curvature one.

Findings

Classified metrics for ${f K}=0$, ${f K}=-1$, and ${f K}=1$ algebraically.

Constructed a new projectively flat Finsler metric with ${f K}=1$ using Minkowskian norms.

Simplified the classification process for spherical symmetric Finsler metrics.

Abstract

In this paper, we study locally projectively flat Finsler metrics with constant flag curvature ${\bf K}$. We prove those are totally determined by their behaviors at the origin by solving some nonlinear PDEs. The classifications when ${\bf K}=0$, ${\bf K}=-1$ and ${\bf K} =1$ are given respectively in an algebraic way. Further, we construct a new projectively flat Finsler metric with flag curvature ${\bf K}=1$ determined by a Minkowskian norm with double square roots at the origin. As an application of our main theorems, we give the classification of locally projectively flat spherical symmetric Finsler metrics much easier than before. ----Comments are welcome.