On a Class of Finsler Metrics of Scalar Flag Curvature

TL;DR

This paper extends the Beltrami Theorem to a broader class of $( ext{alpha},eta)$-metrics in Finsler geometry, establishing conditions under which these metrics are locally projectively flat and classifying those with scalar flag curvature.

Contribution

It generalizes the Beltrami Theorem to $( ext{alpha},eta)$-metrics, including square metrics, and provides conditions for projective flatness and scalar flag curvature classification.

Findings

Beltrami Theorem holds for square metrics in dimension ≥3.

Beta must be closed for the theorem to hold in the broader class.

Classifications of metrics with scalar flag curvature are obtained.

Abstract

We have shown that the Beltrami Theorem in Riemannian geometry is still true for square metrics if the dimension $n\ge 3$, namely, an $n(\ge 3)$-dimensional square metric is locally projectively flat if and only if it is of scalar flag curvature. In this paper, we go on with the study of the Beltrami Theorem for a larger class of $(\alpha,\beta)$-metrics $F=\alpha\phi(\beta/\alpha)$ including square metrics, where $\phi(s)$ is determined by a family of known ODEs satisfied by projectively flat $(\alpha,\beta)$-metrics. For this class, we prove that the Beltrami Theorem holds if $\beta$ is closed, and in particular, we prove that $\beta$ must be closed for a subclass with $\phi(s)$ being a polynomial of degree two. Further, we obtain the local and in part the global classifications to those metrics of scalar flag curvature.