On $(\alpha, \beta, \gamma)$-metrics

TL;DR

This paper introduces a new class of Finsler metrics generalizing $(eta,eta)$-metrics using a Riemannian metric and two 1-forms, and analyzes their conditions for local projective flatness and Douglas type.

Contribution

It defines a novel class of Finsler metrics involving two 1-forms and establishes conditions for their geometric properties.

Findings

Derived necessary and sufficient conditions for local projective flatness.

Established criteria for metrics to be of Douglas type.

Generalized $(eta,eta)$-metrics beyond traditional definitions.

Abstract

In this paper, a new class of Finsler metrics which are included $(\alpha,\beta)$-metrics are introduced. They are defined by a Riemannian metric and two 1-forms $\beta=b_i(x)y^i$ and $\gamma= \gamma_i(x)y^i$. This class of metrics are a generalization of $(\alpha,\beta)$-metrics which are not always $(\alpha,\beta)$-metric. We find a necessary and sufficient condition for this metric to be locally projectively flat and then we prove the conditions for this metric to be of Douglas type.