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

TL;DR

This paper characterizes two-dimensional singular Finsler metrics, especially Kropina and m-Kropina types, identifying conditions for Douglasian and projectively flat cases, revealing unique features of the 2D case compared to higher dimensions.

Contribution

It provides a characterization of two-dimensional singular Finsler metrics, showing their structure and flatness conditions, and highlights differences from higher-dimensional cases.

Findings

Kropina metrics are always Douglasian.

Douglas m-Kropina metrics with m ≠ -1 are locally Minkowskian.

Main class includes m-Kropina plus a linear part on β.

Abstract

Singular Finsler metrics, such as Kropina metrics and $m$-Kropina metrics, have a lot of applications in the real world. In this paper, we study a class of two-dimensional singular Finsler metrics defined by a Riemann metric $\alpha$ and 1-form $\beta$, and we characterize those which are Douglasian or locally projectively flat by some equations. It shows that the main class induced is an $m$-Kropina metric plus a linear part on $\beta$. For this class, the local structure of Douglasian or (in part) projectively flat case is determined, and in particular we show that a Kropina metric is always Douglasian and a Douglas $m$-Kropina metric with $m\ne -1$ is locally Minkowskian. It indicates that the two-dimensional case is quite different from the higher dimensional ones.