Four families of projectively flat Finsler metrics with \(\mathbf{K}=1\) and their non-Riemannian curvature properties (Q1800642)

For the first time, \textit{R. L. Bryant} [Contemp. Math. 196, 27--41 (1996; Zbl 0864.53054)] introduced a locally projectively flat non-Riemannian Finsler metric with flag curvature \(\mathbf K=1\) on \(\mathbb S^2\). The authors here construct four new non-Riemannian families of projectively flat Finsler metrics with flag curvature \(\mathbf K=1\). These metrics belong to the class of generalized 4-th root metrics. The Finsler metrics of this class which are of isotropic flag curvature are characterized and the necessary and sufficient conditions under which a metric in this class is weakly Einstein, are determined. As well, the Finsler metrics of isotropic \(S\)-curvature of almost vanishing \(\Theta\)-curvature are characterized, and the iff condition under which a Finsler metric in this class has vanishing projective Ricci curvature, is determined.