On general (alpha,beta)-metrics with vanishing Douglas curvature

TL;DR

This paper characterizes and constructs a broad class of Finsler metrics called general $(eta,eta)$-metrics with zero Douglas curvature, providing explicit examples and a necessary and sufficient condition for this property.

Contribution

It derives a complete characterization and explicit construction of general $(eta,eta)$-metrics with vanishing Douglas curvature, advancing understanding of Finsler geometry.

Findings

Derived a necessary and sufficient condition for vanishing Douglas curvature.

Solved the key equation to classify all such metrics under certain conditions.

Constructed many new explicit examples of these metrics.

Abstract

In this paper, we study a class of Finsler metrics called general $(\alpha,\beta)$-metrics, which are defined by a Riemannian metric $\alpha$ and a $1$-form $\beta$. We find an equation which is necessary and sufficient condition for such Finsler metric to be a Douglas metric. By solving this equation, we obtain all of general $(\alpha,\beta)$-metrics with vanishing Douglas curvature under certain condition. Many new non-trivial examples are explicitly constructed.