On some non-Riemannian curvature of Minkowskian product Finsler metrics

Let (M1,F1) and (M2,F2) be two Finsler manifolds, Minkowskian product Finsler metric is the Finsler metric F=f(S,T) endowed on the product manifold M=M1×M2, where S=F12, T=F22, f is product function. In this paper, the stretch curvature, mean stretch curvature, χ-curvature and H-curvature of F are derived in terms of the corresponding objects of its components. It turns out that F is a stretch metric or weakly stretch metric if and only if F1 and F2 both are stretch metric or weakly stretch metric. Necessary and sufficient conditions for F to be of vanishing χ-curvature or H-curvature are obtained. Under certain condition, we obtain a differential equation to characterize F with almost vanishing χ-curvature. We also prove that F with almost vanishing H-curvature has vanishing H-curvature.