On Square Metrics of Scalar Flag Curvature

TL;DR

This paper studies square Finsler metrics derived from Riemannian metrics and 1-forms, establishing conditions for projective flatness and scalar flag curvature, and classifying such metrics on closed manifolds in dimensions three and higher.

Contribution

It proves an analogue of the Beltrami Theorem for square metrics and classifies closed manifolds with these metrics of scalar flag curvature in dimensions ≥3.

Findings

Square metrics are locally projectively flat iff they have scalar flag curvature.

The local structure of such metrics is characterized.

Classification of closed manifolds with these metrics is achieved.

Abstract

We consider a special class of Finsler metrics --- square metrics which are defined by a Riemannian metric and a 1-form on a manifold. We show that an analogue of the Beltrami Theorem in Riemannian geometry is still true for square metrics in 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. Further, we determine the local structure of such metrics and classify closed manifolds with a square metric of scalar flag curvature in dimension $n\ge 3$.