On Generalized Douglas-Weyl $(\alpha, \beta)$-Metrics

TL;DR

This paper characterizes generalized Douglas-Weyl $(eta, eta)$-metrics, showing they are Berwald metrics under certain conditions, and explores properties of related Finsler metrics with applications to specific metric types.

Contribution

It establishes a characterization of generalized Douglas-Weyl $(eta, eta)$-metrics as Berwald metrics when not of Randers type and investigates properties of almost regular Finsler metrics.

Findings

Generalized Douglas-Weyl $(eta, eta)$-metrics are Berwald metrics if not of Randers type.

Non-Berwald metrics can be almost regular Finsler metrics that are neither Douglas nor Weyl.

Square and Matsumoto metrics with isotropic mean Berwald curvature are Berwald metrics.

Abstract

In this paper, we study generalized Douglas-Weyl $(\alpha, \beta)$-metrics. Suppose that an regular $(\alpha, \beta)$-metric $F$ is not of Randers type. We prove that $F$ is a generalized Douglas-Weyl metric with vanishing S-curvature if and only if it is a Berwald metric. Moreover by ignoring the regularity, if $F$ is not a Berwald metric then we find a family of almost regular Finsler metrics which is not Douglas nor Weyl. As its application, we show that generalized Douglas-Weyl square metric or Matsumoto metric with isotropic mean Berwald curvature are Berwald metrics.