Some Ricci-Flat $(\alpha,\beta)$-Metrics

TL;DR

This paper investigates Ricci-flat $(eta,eta)$-metrics within Finsler geometry, deriving a key equation that characterizes such metrics when the 1-form's length is constant, advancing understanding of special geometric structures.

Contribution

It introduces a characterization equation for Ricci-flat $(eta,eta)$-metrics with constant length 1-forms, expanding the classification of Ricci-flat Finsler metrics.

Findings

Derived a characterization equation for Ricci-flat $(eta,eta)$-metrics.

Identified conditions under which these metrics are Ricci-flat.

Contributed to the classification of special Finsler geometries.

Abstract

In this paper, we study a special class of Finsler metrics, $(\alpha,\beta)$-metrics, defined by $F = \alpha \phi(\frac{\beta}{\alpha})$, where $\alpha$ is a Riemannian metric and $\beta$ is a 1-form. We find an equation that characterizes Ricci-flat $(\alpha,\beta)$-metrics under the condition that the length of $\beta$ with respect to $\alpha$ is constant.