  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.
