Finsler metrics that are both Douglas and generalized Berwald in dimension two

TL;DR

This paper characterizes two-dimensional Finsler metrics that are both Douglas and generalized Berwald, showing they are either Berwald or specific Randers metrics with closed, constant-length 1-forms.

Contribution

It provides a complete characterization of two-dimensional Finsler metrics that are both Douglas and generalized Berwald, identifying them as either Berwald or certain Randers metrics.

Findings

Finsler metrics in dimension two are Douglas and generalized Berwald iff they are Berwald or Randers with closed, constant-length 1-form.

The characterization is both necessary and sufficient.

The result simplifies the classification of such Finsler metrics in two dimensions.

Abstract

We proof that in dimension two, a Finsler metric is Douglas and generalized Berwald, if and only if it is Berwald or a Randers metric $\alpha + \beta$, where $\beta$ is closed and is of constant length with respect to $\alpha$.