On a class of projectively flat Finsler metrics

TL;DR

This paper classifies a specific class of Finsler metrics called general (α,β)-metrics that are projectively flat when the underlying Riemann metric is projectively flat, leading to the discovery of new projectively flat Finsler metrics.

Contribution

The paper provides a classification of projectively flat general (α,β)-metrics and identifies a new group of such metrics by solving nonlinear PDEs.

Findings

Complete classification of projectively flat general (α,β)-metrics.

Discovery of a new group of projectively flat Finsler metrics.

Solution of nonlinear PDEs characterizing these metrics.

Abstract

In this paper, we study a class of Finsler metrics composed by a Riemann metric $\alpha=\sqrt{a_{ij}(x)y^i y^j}$ and a $1$-form $\beta=b_i(x)y^i$ called general ($\alpha$, $\beta$)-metrics. We classify those projectively flat when $\alpha$ is projectively flat. By solving the corresponding nonlinear PDEs, the metrics in this class are totally determined. Then a new group of projectively flat Finsler metrics is found.