Douglas metrics of (\alpha,\beta) type

TL;DR

This paper classifies (,eta)-metrics with zero Douglas curvature in dimensions three and higher, using -deformations and conformal 1-forms, advancing understanding of Finsler geometry.

Contribution

It introduces -deformations to classify Douglas ,eta)-metrics with vanishing curvature, highlighting the role of conformal 1-forms in their structure.

Findings

Classified (,eta)-metrics with zero Douglas curvature in  dimensions.

Established the importance of conformal 1-forms in these metrics.

Provided a method to construct conformal 1-forms via -deformations.

Abstract

In this paper, the Douglas curvature of (\alpha,\beta)-metrics, a special class of Finsler metrics defined by a Riemannian metric \alpha and a 1-form \beta, is studied. These metrics with vanishing Douglas curvature in dimension n\geq3 are classified by using a new class of metrical deformations called \beta-deformations. The result shows that conformal 1-forms of Riemannian metrics play a key role, and an effective way to construct such 1-forms is provided also by \beta-deformations.