Finsler metrics with constant (or scalar) flag curvature

A Finsler metric on a manifold M with its flag curvature K is said to be almost isotropic flag curvature if K = 3ċ + σ where σ and c are scalar functions on M. In this paper, we establish the intrinsic relation between scalar functions c(x) and σ(x). More general, by invoking the Ricci identities for a one form, we investigate Finsler metric of weakly isotropic flag curvature K = 3θ/F + σ and show that F has constant flag curvature if θ is horizontally parallel.