Finsler manifolds with Positive Weighted Flag Curvature

The flag curvature is a natural extension of the sectional curvature in Riemannian geometry. However there are many non-Riemannian quantities which interact the flag curvature. In this paper, we introduce a notion of weighted flag curvature by modifying the flag curvature with the non-Riemannian quantity, T-curvature. We show that a proper open forward complete Finsler manifold with positive weighted flag curvature is necessarily diffeomorphic to the Euclidean space.