On a Class of Two-Dimensional Einstein Finsler Metrics of Vanishing S-Curvature

TL;DR

This paper investigates a specific class of two-dimensional Finsler metrics defined by Riemannian metrics and 1-forms, characterizing their local structure and showing they are Einsteinian with vanishing S-curvature but not Ricci-flat.

Contribution

It determines the local structure of two-dimensional $(eta)$-metrics with vanishing S-curvature and proves they are Einsteinian but generally not Ricci-flat.

Findings

Metrics are Einsteinian with isotropic flag curvature

Metrics have vanishing S-curvature

Generally not Ricci-flat

Abstract

An $(\alpha,\beta)$-metric is defined by a Riemannian metric $\alpha$ and $1$-form $\beta$. In this paper, we study a known class of two-dimensional $(\alpha,\beta)$-metrics of vanishing S-curvature. We determine the local structure of those metrics and show that those metrics are Einsteinian (equivalently, isotropic flag curvature) but generally are not Ricci-flat.