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arXiv · 1302.4120

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

Guojun Yang
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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.

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Guojun Yang. 2013-02-17. On a Class of Two-Dimensional Singular Douglas and Projectively flat Finsler Metrics. https://arxiv.org/abs/1302.4120

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