A Note on a Class of Finsler Metrics of Isotropic S-Curvature

TL;DR

This paper studies specific Finsler metrics called $(eta)$-metrics with isotropic S-curvature, revealing dimension-dependent classifications and providing explicit examples, especially in two dimensions.

Contribution

It extends the characterization of $(eta)$-metrics with isotropic S-curvature to higher dimensions and identifies additional classes in two dimensions, with explicit constructions.

Findings

Characterization holds for dimensions n≥3.

In 2D, an extra class of isotropic S-curvature exists.

Constructed examples for all 2D classes, including non-constant $eta$ norm.

Abstract

An $(\alpha,\beta)$-metric is defined by a Riemannian metric and $1$-form. In this paper, we investigate the known characterization for $(\alpha,\beta)$-metrics of isotropic S-curvature. We show that such a characterization should hold in dimension $n\ge 3$, and for the 2-dimensional case, there is one more class of isotropic S-curvature than the higher dimensional ones. Further, we construct corresponding examples for every two-dimensional class, especially for the class that the norm of $\beta$ with respect to $\alpha$ is not a constant.