A class of projectively flat Finsler metrics (Q6609584)

This paper deals with a certain class of projectively flat Finsler metrics. In particular, the authors study a class of Finsler metrics of cohomogeneity two on \(\mathbb{R}^{n}\), which are weakly orthogonally invariant Finsler metrics. The authors find conditions for a Finsler metric on \(\mathbb{R}\times\mathbb{R}^{n}\) to be a weakly orthogonally invariant metric which is non-trivial in the sense that this metric is not of orthogonal invariance. The main findings of this paper lie in Theorem 1.1 and Theorem 1.2. In Theorem 1.1, the authors find necessary and sufficient conditions for a weakly orthogonally invariant Finsler metric to be projectively flat. Using this result, they verify that the \(2\)-parameter family of Bryant's metrics \N[\textit{R. L. Bryant}, Houston J. Math. 28, No. 2, 221--262 (2002; Zbl 1027.53086)] is projectively flat with constant flag curvature \(1\). In Theorem 1.2, they obtain conditions for a special Finsler metric to be a locally projectivey flat Finsler metric of scalar flag curvature. Finally, they construct some new projectively flat Finsler metrics on \(\mathbb{S}^{n+1}\) via Bryant's examples and determine their scalarflag curvature.