On the Flag Curvature of Finsler Metrics of Scalar Curvature

TLDR: In this paper, the flag curvature of a Finsler metric is defined as a scalar function on the slit tangent bundle, and the curvature is determined when certain non-Riemannian quantities such as Cartan torsion and Landsberg curvature are isotropic.

Abstract: The flag curvature of a Finsler metric is called a Riemannian quantity because it is an extension of sectional curvature in Riemannian geometry. In Finsler geometry, there are several non-Riemannian quantities such as the (mean) Cartan torsion, the (mean) Landsberg curvature and the S-curvature, which all vanish for Riemannian metrics. It is important to understand the geometric meanings of these quantities. In this paper, we study Finsler metrics of scalar curvature (i.e., the flag curvature is a scalar function on the slit tangent bundle) and partially determine the flag curvature when certain non-Riemannian quantities are isotropic. Using the obtained formula for the flag curvature, we classify locally projectively flat Randers metrics with isotropic S-curvature.