On pseudo-Finsler manifolds of scalar flag curvature

Let (M,L)(M,L) be a pseudo-Finsler manifold, ξξ the geodesic spray vector field associated to the non-degenerate, 2-positively homogeneous Lagrangian LL. In this paper we prove that (M,L)(M,L) is of scalar flag curvature kk if and only if the equation Lξg+kλLξgˆ=0 holds on Γ(IλM)Γ(IλM), the Lie algebra of tangent vector fields to the λλ-indicatrix bundle IλMIλM, where gg and gˆ are pseudo-Riemannian metrics on the vertical and respectively on the horizontal subbundle. Also, we prove that any pseudo-Finsler manifold is of scalar flag curvature at any point of the light cone.