[Paper Review] On a Class of Singular Projectively Flat Finsler Metrics with Constant Flag Curvature

Guojun Yang|arXiv (Cornell University)|Feb 14, 2013

Advanced Differential Geometry Research13 references3 citations

TL;DR

This paper classifies singular $(\alpha,\beta)$-metrics that are locally projectively flat with constant flag curvature in dimensions $n=2$ and $n\geq 3$. It proves that such metrics are either flat-parallel or locally Minkowskian, with explicit constructions for $m$-Kropina and Kropina metrics, and shows that constant flag curvature does not imply flat-parallelism, extending known results on Randers and $(\alpha+\beta)^2/\alpha$ 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 classify a class of singular $(α,β)$-metrics which are locally projectively flat with constant flag curvature in dimension $n= 2$ and $n \ge 3$ respectively. Further, we determine the local structure of $m$-Kropina metrics and particularly Kropina metrics which are projectively flat with constant flag curvature and prove that such metrics must be locally Minkowskian but are not necessarily flat-parallel.

Motivation & Objective

• To classify singular $(\alpha,\beta)$-metrics that are locally projectively flat with constant flag curvature in dimensions $n=2$ and $n\geq 3$.
• To determine the local structure of $m$-Kropina and Kropina metrics under these curvature and projective flatness conditions.
• To extend known classifications of regular $(\alpha,\beta)$-metrics with constant flag curvature to the singular case where $\phi(0)=0$ or undefined.
• To show that such metrics are locally Minkowskian but not necessarily flat-parallel, resolving a key distinction from the regular case.

Proposed method

• Analyzes $(\alpha,\beta)$-metrics of the form $F = \alpha\phi(s)$, $s=\beta/\alpha$, with $\phi(s) = cs + s^m\varphi(s)$, $m \neq 0,1$, $\varphi(0)=1$, and $c=0$ if $m$ is a negative integer.
• Applies the projective flatness condition via the projective factor $P$ and derives the associated system of PDEs for $\alpha$ and $\beta$.
• Uses a deformation $\widetilde{\alpha} = \alpha/b$, $\widetilde{\beta} = \beta/b^2$ to simplify the metric structure and reduce to flatness and parallelism conditions.
• Solves the resulting ODEs and PDEs for the coefficients $r_{00}$, $s_{ij}$, and $G^i_\alpha$, leading to classification into flat-parallel or specific $m$-Kropina forms.
• Employs Taylor expansions and analyticity assumptions to derive explicit local expressions for $\alpha$ and $\beta$ in the constant flag curvature case.
• Applies the Beltrami and Schur theorems in the context of singular metrics to constrain curvature and projective geometry.

Experimental results

Research questions

• RQ1 Under what conditions is a singular $(\alpha,\beta)$-metric with $\phi(s) = cs + s^m\varphi(s)$ locally projectively flat with constant flag curvature?
• RQ2 Can $m$-Kropina and Kropina metrics ($m=-1$) be projectively flat with constant flag curvature without being flat-parallel?
• RQ3 What are the explicit local forms of $\alpha$ and $\beta$ for such metrics in dimensions $n=2$ and $n\geq 3$?
• RQ4 How does the constant flag curvature condition constrain the geometry of singular $(\alpha,\beta)$-metrics beyond the regular case?
• RQ5 Is the class of projectively flat singular $(\alpha,\beta)$-metrics with constant flag curvature strictly larger than the flat-parallel or known regular cases?

Key findings

• For $n \geq 2$, a singular $(\alpha,\beta)$-metric $F = \alpha\phi(s)$ with $\phi(s) = cs + s^m\varphi(s)$ is projectively flat with constant flag curvature if and only if it is flat-parallel, i.e., $\alpha$ is flat and $\beta$ is parallel with respect to $\alpha$.
• For $n \geq 2$, the metric $F = \beta^m(\alpha^2 + k\beta^2)^{(1-m)/2}$ is projectively flat with zero flag curvature $K=0$, and is locally Minkowskian but not necessarily flat-parallel.
• For $n=2$, the metric $F = \frac{2b}{1-kb^2}\left\{b\sqrt{\alpha^2 - k\beta^2} - \sqrt{b^2\alpha^2 - \beta^2}\right\}$ is projectively flat with negative constant flag curvature $K<0$, where $b = \|\beta\|_\alpha$.
• The $m$-Kropina metric $F = \beta^m \alpha^{1-m}$ is projectively flat with constant flag curvature if and only if it is locally Minkowskian, but not necessarily flat-parallel, as shown via deformation to $\widetilde{\alpha}$ and $\widetilde{\beta}$.
• For the fifth class $F = k_1\beta + 2k_2\alpha^2/\beta + \alpha^4/\beta^3$ with $k_1 - k_2^2 \neq 0$, projective flatness with constant flag curvature implies $K=0$ and $F$ is locally Minkowskian with $\alpha$ flat and $\beta$ parallel.
• The paper proves that constant flag curvature does not imply flat-parallelism for singular metrics, even when the metric is locally Minkowskian, thus distinguishing the singular case from the regular one.