On a Class of Finsler Metrics of Scalar Flag Curvature
Guojun Yang

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 , namely, an -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 -metrics including square metrics, where is determined by a family of known ODEs satisfied by projectively flat -metrics. For this class, we prove that the Beltrami Theorem holds if is closed, and in particular, we prove that must be closed for a subclass with being a polynomial of degree two. Further, we obtain the local and in part the global classifications to those metrics of scalar flag curvature.
Peer Reviews
No public reviews on file for this paper yet. If you reviewed it on a platform where reviews are public (OpenReview, ICLR, NeurIPS, ICML), you can paste yours below so the community can read it here.
Videos
No videos yet. Explain this paper in a talk, walkthrough, or lecture? Add one.
Taxonomy
TopicsAdvanced Differential Geometry Research · Geometric Analysis and Curvature Flows
