Four families of projectively flat Finsler metrics with K=1 and their non-Riemannian curvature properties

For the first time, Bryant introduced a locally projectively flat non- Riemannian Finsler metric with flag curvature K = 1 on S2. In this paper, we construct four new nonRiemannian families of projectively flat Finsler metrics with flag curvature K = 1. These metrics belong to the class of generalized 4- th root metrics. The class of generalized 4- th root metrics is an extension of the class of 4- th root metrics which has close relations with General Relativity and Seismic Ray Theory. In this class, we characterize Finsler metrics of isotropic flag curvature. We find the necessary and sufficient condition under which a metric in this class be weakly Einstein. In this class, we characterize Finsler metrics of isotropic S- curvature and almost vanishing - curvature. Finally, we find the necessary and sufficient condition under which a Finsler metric in this class has vanishing projective Ricci curvature.