On a Class of Two-Dimensional Douglas and Projectively Flat Finsler Metrics

TL;DR

This paper investigates a specific class of two-dimensional Finsler metrics, characterizing when they are Douglasian or locally projectively flat, and reveals new properties about the 1-form etaorm in two dimensions.

Contribution

It provides a characterization of two-dimensional (lpha,etaorm) Finsler metrics that are Douglasian or projectively flat, including cases where etaorm is not closed, which contrasts with higher-dimensional results.

Findings

etaorm is not necessarily closed in two dimensions.

Explicit local structures for Douglasian (lpha,etaorm) metrics are determined.

Examples of projectively flat metrics with non-closed etaorm are provided.

Abstract

In this paper, we study a class of two-dimensional Finsler metrics defined by a Riemannian metric $\alpha$ and a 1-form $\beta$. We characterize those metrics which are Douglasian or locally projectively flat by some equations. In particular, it shows that the known fact that $\beta$ is always closed for those metrics in higher dimensions is no longer true in two dimensional case. Further, we determine the local structures of two-dimensional $(\alpha,\beta)$-metrics which are Douglassian, and some families of examples are given for projectively flat classes with $\beta$ being not closed.