On conformally flat (α,β)-metrics with relatively isotropic mean Landsberg curvature

In this paper, we study conformally flat (alpha, beta)-metrics in the form of F = alpha phi(beta/alpha), where alpha is a Riemannian metric and beta is a 1-form on the manifold. We prove that conformally flat weak Landsberg (alpha, beta)-metrics must be either Riemannian metrics or locally Minkowski metrics. Further, we prove that, if phi(s) is a polynomial in s, then conformally flat (alpha, beta)-metrics with relatively isotropic mean Landsberg curvature must also be either Riemannian metrics or locally Minkowski metrics.