Finsler manifolds with Positive Weighted Flag Curvature

Abstract

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