On Square Metrics of Scalar Flag Curvature

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 $n\ge 3$.