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

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.

21 pages. arXiv admin note: substantial text overlap with arXiv:1302.3150, arXiv:1302.3300

References in corpus (1)

• On a Class of Singular Projectively Flat Finsler Metrics with Constant Flag Curvature

Cited by in corpus (2)

• On a Class of Singular Projectively Flat Finsler Metrics with Constant Flag Curvature
• On a Class of Singular Douglas and Projectively flat Finsler Metrics