Projectively flat general $(\alpha,\beta)$-metrics with constant flag curvature

TL;DR

This paper classifies a specific class of Finsler metrics called general $(eta)$-metrics with constant flag curvature, identifying conditions for local projective flatness and constructing new examples with various curvature values.

Contribution

It provides a complete classification of general $(eta)$-metrics with constant flag curvature under certain conditions, including the construction of new singular projectively flat metrics.

Findings

Classified general $(eta)$-metrics with constant flag curvature

Constructed new projectively flat Finsler metrics with curvatures 1, 0, -1

Identified singularities in the constructed metrics

Abstract

In this paper we study the flag curvature of a particular class of Finsler metrics called general $(\alpha,\beta)$-metrics, which are defined by a Riemannian metric $\alpha$ and a $1$-form $\beta$. The classification of such metrics with constant flag curvature are completely determined under some suitable conditions, which make them be locally projectively flat. As a result, we construct some new projectively flat Finsler metrics with flag curvature $1$, $0$ and $-1$, all of which are of singularity at some directions.